KYIV NATIONAL TARAS SHEVCHENKO UNIVERSITY INTERNATIONAL CONFERENCE MODERN STOCHASTICS: THEORY AND APPLICATIONS Dedicated to the 60th anniversary of the Department of Probability Theory and Mathematical Statistics and to the memory of Professor M.Y. Yadrenko (1932-2004) CONFERENCE MATERIALS June 19-23, 2006 МIЖНАРОДНА КОНФЕРЕНЦIЯ СУЧАСНА СТОХАСТИКА: ТЕОРIЯ I ЗАСТОСУВАННЯ Присвячена 60-рiччю кафедри теорiї ймовiрностей i математичної статистики та пам’ятi М.Й.Ядренка (1932-2004) МАТЕРIАЛИ КОНФЕРЕНЦIЇ 19-23 червня, 2006 KYIV 2006
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KYIV NATIONAL TARAS SHEVCHENKO UNIVERSITY
INTERNATIONAL CONFERENCEMODERN STOCHASTICS:
THEORY AND APPLICATIONS
Dedicated to the 60th anniversary of the Department of ProbabilityTheory and Mathematical Statistics and to the memory of
Professor M.Y. Yadrenko (1932-2004)
CONFERENCE MATERIALS
June 19-23, 2006
МIЖНАРОДНА КОНФЕРЕНЦIЯСУЧАСНА СТОХАСТИКА:
ТЕОРIЯ I ЗАСТОСУВАННЯ
Присвячена 60-рiччю кафедри теорiї ймовiрностей i математичноїстатистики та пам’ятi М.Й.Ядренка (1932-2004)
МАТЕРIАЛИ КОНФЕРЕНЦIЇ
19-23 червня, 2006
KYIV 2006
PROGRAMME COMMITTEE
V. Buldygin, A. Gushchin, O. Ivanov, N. Kartashov, P. Knopov,Yu. Kondratiev, V. Korolyuk, Yu. Kozachenko, O. Kukush, G. Kulinich,N. Leonenko, R. Maiboroda, S. Makhno, A. Nakonechnyi, M. Portenko,M. Pratsiovytyi, D. Silvestrov, A. Skorohod, N. Zinchenko.
ORGANIZING COMMITEE
O. Zakusilo (Chairman), Yu. Mishura (Vice-chairman), I. Parasyuk,M. Perestyuk, O. Borisenko, O. Kurchenko, M. Moklyachuk, A. Olenko,O. Ponomarenko, L. Sakhno (Scientific secretary), G. Bagro (Secretary),N. Semenovska (Secretary), G. Shevchenko (Secretary), O. Vasylyk (Secretary),T. Yakovenko (Secretary), R. Yamnenko (Secretary).
ПРОГРАМНИЙ КОМIТЕТ
В. Булдигiн, А. Гущiн, О. Iванов, Н. Зiнченко, М. Карташов, П. Кнопов,Ю. Козаченко, Ю. Кондратьєв, В. Королюк, О. Кукуш, Г. Кулiнiч,М. Леоненко, Р. Майборода, С. Махно, О. Наконечний, М. Портенко,М. Працьовитий, Д. Сiльвестров, А. Скороход.
ОРГАНIЗАЦIЙНИЙ КОМIТЕТ
О. Закусило (голова), Ю. Мiшура (заступник голови), I. Парасюк,М. Перестюк, О. Борисенко, О. Курченко, М. Моклячук, А. Оленко,О. Пономаренко, Л. Сахно (вчений секретар), Г. Багро (секретар), О. Василик(секретар), Н. Семеновська (секретар), Г. Шевченко (секретар), Т. Яковенко(секретар), Р. Ямненко (секретар).
Department of Probability Theory and MathematicalStatistics at Kyiv National University
Prehistory of Probability Theory at Kyiv National University
The first lecture on probability theory at Kyiv University was given in 1863by Mikhail Vashchenko-Zakharchenko (1825 – 1912). The first textbook in proba-bility theory written by professor Vasyl’ Ermakov (1845 – 1922) was published in“Proceedings of the University” in 1878. At that time it was one of the most up-to-date books. In particular, it included new results on the law of large numbersobtained by P. Chebyshev. In 1905 there was published the book by V. Ermakov“The Least Squares Method”.
Professor D. Grave (1863 – 1939) was the first lecturer in actuarial mathemat-ics. He published the textbooks: “Mathematics of Insurance” (1912), “Theory ofPension Funds” (1917), “Mathematics of Social Insurance” (1924). Thus, togetherwith University of Stockholm (F. Lundberg, H. Cramer) Kyiv University was thepioneer in creation and development of actuarial mathematics.
It is worth to recall the eminent scientist Eugene Slutskyi (1880 – 1948),who was a founder of mathematical economics and theory of stochastic processes(together with A. Khinchin and A. Kolmogorov). E. Slutskyi is one of the famousgraduates of Kyiv University.
In the late twenties and early thirties of the twentieth century academicianMikhail Kravchuk (1892 – 1942) developed probability-theoretical and statisticalresearch at Kyiv University. M. Kravchuk and his disciples O. Smogorzhevskyi(1896 – 1969), K. Latysheva (1897 – 1956), O. Lebedyntseva, S. Kulyk studiedorthogonal polynomials corresponding to some probability distributions. Nowa-days, polynomials generated by Bernoulli distribution bear Kravchuk’s name.
In 1934 Mykola (Nikolai) Bogolyubov (1909 – 1992) started to work at theUniversity. At that time M. Bogolyubov and M. Krylov (1879 – 1955) publishedtheir classical papers on ergodic theory of Markov chains with an arbitrary phasespace. The paper “On Fokker–Planck equations derived in the theory of perturba-tion by the method based on spectral properties of the perturbation Hamiltonian”played very important role in the development of theory of stochastic processes.Mathematical substantiation of this paper resulted in creation of the theory of sto-chastic differential equations which is one of the most important parts of moderntheory of stochastic processes.
In 1939 – 1941 Bogolyubov’s post-graduate student Iosyp Gikhman (1918 –1985) studied dynamic systems influenced by random forces, striving to substan-tiate the limit transitions. Just before World War II he had prepared candidate’s(Ph.D.) thesis “Dynamic systems under the influence of random forces”. Duringthe war Gikhman was in the army in the field. In 1942, during a short leave fromthe army, he defended his Ph.D. thesis at the Institute of Mathematics of Uzbek-istan Academy of Sciences. After the war 1945 I. Gikhman worked at the Roadsand Vehicle Institute (1945 – 47) and at Kyiv University (since 1947). He focusedon the rigorous definition of stochastic differential equation and conditions for
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existence of its solutions. Concurrently, K. Ito was studying similar problems faraway in Japan.
According to archival documents, in 1945 Mechanics and Mathematics Fac-ulty of Kyiv University consisted of seven departments: mathematical analysis,geometry, algebra and probability theory, elasticity theory, mathematical physicsand function theory, engineering mechanics, general mathematics. Department ofAlgebra and Probability Theory was headed by associated professor M. Sokolov.S. Avramenko, O. Lebedyntseva and I. Gikhman worked at the Department.
In 1949 academician of the Ukrainian Academy of Sciences Borys Gnedenkobecame a Head of Department of Algebra and Probability Theory at Kyiv Univer-sity. B. Gnedenko was a disciple of A. Kolmogorov and A. Khinchin and broughtto Kyiv University the best traditions and spirit of the Moscow mathematicalschool. This school featured with the support of aspiration of young mathemati-cians to carry out independent research and also in the involving of young peoplein scientific work. No wonder, B. Gnedenko attracted a lot of young scientists.Among his first students were future academicians V. Korolyuk, V. Mykhalevych,A. Skorokhod.
In 1949 the “Radyanska Shkola” publishing house issued the first Ukrainianlanguage edition of the textbook “The Course of Probability Theory” by B. Gne-denko. One can hardly recall another textbook, which have run through so manyeditions in several languages: 8 editions in Russian, 11 in German, 6 in English.Also the textbook was translated in Hungarian, Polish, Japanese, Chinese, French,Italian, Vietnamese and Arabic languages.
In 1955, when the Head of Department of Mathematical Analysis G. Shylovmoved to Moscow, this department was united with Department of Algebraand Probability Theory into Department of Mathematical Analysis, ProbabilityTheory and Algebra. In 1956 some new scientists joined the department: pro-fessor L. Kaluzhnin, senior lecturers V. Mykhalevych and A. Skorokhod. Also,V. Glushkov, O. Parasyuk, V. Koroluuk and K. Yushchenko (Rvachova) workedat the department.
At the beginning of 1958 Borys Gnedenko managed to overcome numerousbureaucratic obstacles and got a permission to create the specialization “Proba-bility Theory and Mathematical Statistics” at the Mechanics and MathematicsFaculty of Kyiv University. Students specializing in probability theory attendedmandatory courses “Additional topics on probability theory”, “Theory of stochasticprocesses”, “Mathematical statistics”.
In 1958 Borys Gnedenko stopped his work at Kyiv University, but he alwaysmaintained close relations with Department of Probability Theory of Kyiv Uni-versity. Shortly before his death in 1993 Borys Gnedenko wrote: “I am happythat destiny committed me with the honourable mission to found probability-theoretical school in Ukraine. Now, a lot of my scientific grandchildren and great-grandchildren have joined this school. It has won a serious position in the life ofmodern probability theory. It is very important for it to gain not only in quan-tity, but in quality, to discover new fields of research and to keep in touch withpractice”.
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After Borys Gnedenko left Kyiv University, Department of MathematicalAnalysis and Probability Theory was headed by O. Parasyuk, and then byV. Dzyadyk.
In 1962 the separate Department of Probability Theory and MathematicalStatistics was created and professor I. Gikhman became its head. The teachingstaff of the new department consisted of associated professor A. Skorokhod, se-nior lecturer M. Yadrenko, assistant professors A. Dorogovtsev, M. Slobodenyuk,V. Baklan, I. Ezhov. Number of post-graduate students increased, there werefirst foreign post-graduate students – H. Hekendorff (Germany), Li Shi Lin’, Gan’Chzhan Zjuan’ (Chinese People Republic).
In 1961 Kyiv University Press published A. Skorokhod’s monograph “Researchin the Theory of Stochastic Processes” which became a basis for his doctoralthesis defended in 1963. In 1965 this monograph was translated into English inUSA and run through two editions (in 1970, 1982). The monograph imparteda powerful impetus to the development of the theory of stochastic processes inthe sixties – nineties. The notions of “Skorokhod’s method of common probabilityspace”, “Skorokhod’s representation” (of a sequence of sums of independent randomvariables by values of the Wiener process), “Skorokhod’s topology” are often usedin modern scientific literature.
In 1964 I. Gikhman and A. Skorokhod accomplished their “Introduction toTheory of Stochastic Processes” – the first textbook in this field written in Russian.It run through two editions (1965, 1978) and was translated into English, French,Polish, German, Hungarian and Chinese.
It’s impossible to overestimate Skorokhod’s role in the development of prob-ability theory at Kyiv University. The list of monographs written by him im-presses: “Research in Theory of Stochastic Processes” (1961), “Random Processeswith Independent Increments” (1964; revised edition – 1986), “Theory of Sto-chastic Processes” (co-author I. Gikhman; vol.1, 1971; vol.2, 1973; vol.3, 1975),“Stochastic Differential Equations” (co-author I. Gikhman; 1968), “Limit The-orems for Random Walks” (co-author M. Slobodenjuk, 1970), “Integration inHilbert Space” (1975), “Controlled Random Processes” (co-author I. Gikhman,1977), “Random Linear Operators” (1978), “Consistent Estimates of Parametersof Random Processes” (co-author I. Ibramkhalilov, 1980), “Stochastic Differen-tial Equations and their Applications” (co-author I. Gikhman, 1982), “StochasticEquations for Complex Systems” (1983), “Asymptotic Methods in the Theory ofStochastic Differential Equations” (1987). The majority of mentioned monographswas translated in English.
A. Skorokhod had not been abroad until 1983. This circumstance and a vast ofmonographs signed by this name gave rise to the hypothesis, spread among foreignmathematicians, that “A. Skorokhod is a pseudonym of a collective of Ukrainianmathematicians studying stochastic processes”.
In 1965 I. Gikhman was elected a Corresponding Member of the Academy ofSciences of Ukraine and moved to Donetsk where he was appointed the head ofdepartments of probability theory, both at the new Institute for Applied Mathe-matics and Mechanics and at Donetsk University. I. Gikhman had many disciples.
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Many generations of Kyiv University graduates, mathematicians of different spe-cialities, recall warmly his wonderful lectures in mathematical analysis, probabilitytheory, functional analysis, integral equations, mathematical physics,number the-ory, as well as his probabilistic and statistical special courses. Under his guidanceM. Yadrenko, A. Dorogovtsev, I. Ezhov, H. Hekenforff, V. Baklan, M. Portenko,O. Illyashenko, B. Kucher, Li Shi Lin’, H. Mansurov and A. Shatashvili defendedtheir Ph.D. theses.
Department of Probability Theory and Mathematical Statistics:1966 – 2006
The subsequent heads of Department of Probability Theory and Mathemat-ical Statistics were: Corresponding Member of National Academy of Sciences ofUkraine Professor Mikhail Yadrenko (January 1966 – December 1998), ProfessorYuriy Kozachenko (December 1998 – December 2003). The present (since Decem-ber 2003) head is Professor Yuliya Mishura.
In different periods the following scientists worked at the Department: A. Doro-govtsev, I. Ezhov, Yu. Ryzhov, V. Baklan, V. Bandura, G. Pryzva, G. Kulinych,Yu. Kozachenko, D. Silvestrov, M. Kartashov, M. Leonenko, O. Ponomarenko,R. Maiboroda, O. Borysenko, M. Moklyachuk, A. Olenko. Also, A. Skorokhod,V. Korolyuk, V. Shurenkov and M. Portenko gave lectures and were scientificadvisors for post-graduate students at the Department as part-time employees.
All these years the Department has been intensively working on preparationof textbooks and tutorials. In particular, the following textbooks were published:I. Gikhman, A. Skorokhod, M. Yadrenko “Probability Theory and MathematicalStatistics” (1979, 1988), A. Dorogovtsev, D. Silvestrov, A. Skorokhod, M. Ya-drenko “Selected Problems in Probability Theory” (1976, 1980; English transla-tion – 1998), A. Skorokhod “Lectures in Theory of Stochastic Processes” (1990),A.Skorokhod “Elements of Probability Theory and Theory of Stochastic Processes”(1975).
The Department has considerably extended the gamut of scientific research.At present, the main trends of investigation are:
• limit theorems of the theory of stochastic processes;
• theory of random fields;
• statistics of random processes and fields;
• Markov processes; semi-Markov processes;
• stochastic differential equations;
• queueing theory and reliability theory;
• Gaussian random processes and their generalizations;
• stochastic analysis;
• actuarial and financial mathematics.
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More than 150 post-graduates defended their Ph.D. theses. Many of themdefended afterwards doctoral theses. Now they are prominent scientists and leadimportant scientific divisions (in particular, at Kyiv University).
The Department of Probability Theory and Mathematical Statistics has be-come one of the leading organizations in development of the theory of randomfields. The research in this area was summarized in the monographs “SpectralTheory of Random Fields” (1980; English translation – 1983) by M. Yadrenko ,“Statistical Analysis of Random Fields” (1986) by M. Leonenko and O. Ivanov,“Limit Theorems for Random Fields with Singular Spectrum” (1999) by M. Leo-nenko.
Most representatives of the world-wide recognized Ukrainian school of prob-ability theory graduated from the Department. Seven Full Members and eightCorresponding Members of National Academy of Sciences, 40 Doctors of Sciencesare its graduates. Allumni of the Department work at research and educationalinstitutions in 27 countries.
At present, the teaching staff of the Department consists of:Mishura Yuliya – head of department, professor, doctor of science in physics
and mathematics (functional limit theorems for random fields; martingale theory;stochastic integration; stochastic calculus with fractional Brownian motion; finan-cial mathematics; theory of multiparameter random processes);
Borysenko Oleksandr – associate professor, candidate of science in physicsand mathematics (stochastic differential equations’ theory, actuarial and financialmathematics, computer simulation in financial mathematics, computer statistics);
Kartashov Mykola – professor, doctor of science in physics and mathematics(Markov processes, queueing theory, random processes in actuarial mathematics,statistical problems in demography, computer statistics);
Kozachenko Yuriy – professor, doctor of science in physics and mathematics(random processes and fields with values in some functional spaces, statisticalsimulation of random processes and fields, applied statistics);
Maiboroda Rostyslav – professor, doctor of science in physics and mathe-matics (statistical analysis of random processes and fields, applied statistics);
Moklyachuk Mikhail – professor, doctor of science in physics and mathe-matics (theory of random processes and fields, optimization problems in the theoryof stochastic processes, modelling in financial mathematics, applied statistics);
Olenko Andrew – associate professor, candidate of science in physics andmathematics (statistics of random processes and fields, time series and their ap-plications in econometrics, financial mathematics, modelling in financial and ac-tuarial mathematics, computer statistics);
Ponomarenko Oleksandr – associate professor, candidate of science inphysics and mathematics (theory of random fields, general stochastic analysis,mathematical economics and econometrics, actuarial and financial mathematics,random processes in social sciences);
Vasylyk Olga – assistant professor, candidate of science in physics and math-ematics (random processes and fields with values in some functional spaces, sta-
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tistical simulation of random processes and fields, methods of survey sampling);Yamnenko Rostyslav – assistant professor (random processes and fields
with values in some functional spaces, statistical simulation of random processesand fields, generalized fractional Brownian motion, queueing theory);
Bagro Galyna – senior laboratorian (organizational and publishing activity);Lapida Tetyana – senior laboratorian (organizational activity concerned
with teaching process).Besides that, the following scientists work at the Department as part-time
employees:Gusak Dmytro – professor, doctor of science in physics and mathematics
(boundary problems in theory of stochastic processes, risk processes in actuarialmathematics);
Masol Volodymyr – professor, doctor of science in physics and mathemat-ics (limit theorems for solutions of Boolean equations with random coefficients,cryptography, information protection);
Portenko Mykola – Corresponding Member of National Academy of Sci-ences of Ukraine, professor, doctor of science in physics and mathematics (theoryof stochastic processes, diffusion processes).
The scientific-research staff of the Department consists of:Zinchenko Nadiya – leading scientific associate, doctor of sciences in physics
and mathematics (limit theorems in theory of stochastic processes and fields,stable distributions and their application, risk theory);
Sakhno Ludmila – senior scientific associate, candidate of science in physicsand mathematics (limit theorems in theory of random processes and fields, statis-tical analysis of random processes and fields);
Masol Victoria – junior scientific associate, candidate of sciences in physicsand mathematics (cryptography, mathematical methods for information protec-tion, risk theory, actuarial mathematics);
Yakovenko Tetyana – engineer, candidate of science in physics and mathe-matics (theory of stochastic processes);
Semenovska Natalia – engineer (theory of stochastic processes);Klokun Svitlana – senior laboratorian.
Educational activityThe Department of Probability Theory and Mathematical Statistics of Kiev
University is a world-wide known center of high level specialists and scientiststraining in probability theory, theory of stochastic processes, mathematical statis-tics and applications of these disciplines. For fifty years the Department has beenpreparing qualified specialists in probability theory and mathematical statisticswithin the framework of speciality “Mathematics”.
In 1995 the Department put forward the initiative to prepare mathemati-cally educated statisticians (or mathematicians specializing in statistics). Since1996/97 academic year the Mechanics and Mathematics Faculty of Kyiv NationalUniversity trains bachelors, specialists and masters in the speciality “Statistics”
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on the base of Department of Probability Theory and Mathematical Statistics.The educational program is a standard one for the famous universities teachingin statistics. Within the framework of this speciality the following specializationwere developed :
• actuarial and financial mathematics;
• econometrics and mathematical economics;
• applied statistics.
During last few years teachers prepared and published numerous textbooksand tutorials for new statistical specializations. Some of them are listed below.
• Borysenko O. Maiboroda R. “Analytic Methods in Psychology and Sociol-ogy”, Kiev University, 1999.
• Borysenko O., Mishura Yu., Radchenko V., Vasylyk O. “Collected Problemsin Financial Mathematics ”, Kyiv University, 2005.
• Kartashov M. “Strong Stable Markov Chains”, VSP, 1996.
• Kartashov M. “Probability Theory and Mathematical Statistics”, Kyiv:TBiMC, 2004.
• Kozachenko Yu., Pashko A. “Simulation of Random Processes”, Kyiv Uni-versity, 1999.
• Buldygin V., Kozachenko Yu. “Metric Characteristics of Random Variablesand Processes”, Kyiv: TBiMC, 1998 (In Russian). English translation: AMS,Providence RI, 2000.
• Kozachenko Yu. “Lectures in Wavelet Analysis”, Kyiv: TBiMC, 2004.
• Leonenko, M., Mishura, Yu., Parkhomenko, V., Yadrenko, M. “Probabilisticand Statistical Methods in Econometrics, Actuarial and Financial Mathe-matics”, Kyiv, 1995.
• Maiboroda R. “Computational Statistics”, Kyiv University, 2002.
• Maiboroda R. “Statistical Analysis of Mixtures”, Kyiv University, 2002.
• Moklyachuk M. “Calculus of Variations. Extremum Problems”, Kyiv: “Ly-bid” ’, 1994; Kyiv: VPTS “Ekspress”, 2003; Moskva: R&C Dynamics, 2005.
• Moklyachuk M. “Basics of Convex Analysis”, Kyiv: TBiMC, 2004.
• Olenko A., Yadrenko M. “Discrete Mathematics”, Kyiv University, 1997.
• Olenko A., Zinchenko N. “Analytical Models and Methods in Social Sci-ences”, 1998.
• Olenko A. “Actuarial Mathematics. Problems”, Kyiv University, 2005.
• Parkhomenko V. “Methods of Sample Surveys”, Kyiv, 2001.
• Ponomarenko O., Ponomarenko V. “System Methods in Economics, Man-agement and Business”, Kyiv: “Lybid” ’, 1995.
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• Ponomarenko O., Perestyuk M., Burym V. “Foundations of MathematicalEconomics”, Kyiv: “Informtechnika”, 1995.
• Ponomarenko O. “Modern Analytical Politology”, Kyiv University, 2000.
• Ponomarenko O. “Foundations of Financial Mathematics and Insurance”,Kyiv: IPC of State Statistics Committee of Ukraine, 2004.
• Ponomarenko O., Perestyuk M., Burym V. “Modern Economic Analysis.Macroeconomics”, Kyiv: “Vyshcha Shkola”, 2001.
• Ponomarenko O., Perestyuk M., Burym V. “Modern Economic Analysis.Microeconomics”, Kyiv: “Vyshcha Shkola”, 2001.
• Yadrenko M. “Discrete Mathematics”, Kyiv: Ekspress, 2003.
In last five years the Department received several International Grants for edu-cational projects in the field of statistical education. This afforded an opportunityto organize modern material and technical basis for providing teaching process inspeciality “Statistics”, to create library of scientific and training literature, and tointensify process of preparation and publishing new textbooks and tutorials.
Scientific Research
Besides the traditional trends originated by B. Gnedenko, I. Gikhman, V. Ko-rolyuk, A. Skorokhod and M. Yadrenko and listed above the Department developsnew ones. In the last decade, a good deal of the research projects at the Depart-ment are devoted to the applied investigations in the national economy problems.Since 1980 the professors and fellows of the Department have been elaboratingmethods, algorithms and program systems for probabilistic and statistical analysisin the following fields:
• meteorological forecast and ecology;
• processing of trajectory and telemetric information in space investigation;
• reliability analysis of technical products;
• stratification of geological layers;
• analysis of psychological data;
• modelling of stochastic fields; processing and transmission of data.
The staff of the Department takes part in the following researches:
• profound statistical analysis of economic and ecological information, dataof psychological and sociological investigations;
• elaboration of forecast methods, classification based on modern theoreticallyprobabilistic methods;
• elaboration of probabilistic mathematical models in econometric and socio-metric investigations;
• method elaboration of data storing and defense.
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Students of Kyiv University specializing in probability theory and mathemati-cal statistics actively participate in the research at the Department. They write thelong-term projects, participate in scientific seminars, optional study and researchgroups, hold student conferences, scientific meetings and days of science.
The Department of Probability Theory and Mathematical Statistics was aco-organizer of international schools and conferences:
• The Eighth International School on Mathematical and Statistical Methodsin Economics, Finance and Insurance (June, 2004, Foros, Ukraine)
• The Seventh International School on Mathematical and Statistical Methodsin Economics, Finance and Insurance (September 2003, Laspi, Ukraine)
• International Gnedenko Conference (June 2002, Kyiv, Ukraine)
• The Sixth International School on Applied Statistics, Actuarial and Finan-cial Mathematics (September 2002, Laspi, Ukraine)
• The Fifth International School on Applied Statistics, Actuarial and Finan-cial Mathematics (June 2001, Gurzuf, Ukraine)
• The Fourth International School on Applied Statistics, Actuarial and Fi-nancial Mathematics (January 2001, Vasteras, Sweden)
• The Third International School on Applied Statistics, Actuarial and Finan-cial Mathematics (September 2000, Feodosiya, Ukraine)
• The Third Scandinavian–Ukrainian Conference on Probability and Statis-tics (June 1999, Kyiv, Ukraine)
• The Second International School on Actuarial and Financial Mathematics(June 1999, Kyiv, Ukraine)
• The First International School on Financial Mathematics and MathematicalEconomics (September 1998, Kyiv, Ukraine)
• The Second Scandinavian–Ukrainian Conference on Probability and Statis-tics (1997, Umea, Sweden)
• The First Scandinavian–Ukrainian Conference on Probability and Statistics(1995, Uzhgorod, Ukraine)
• Ukrainian–Hungarian Conference on Probability Theory (1993, Mukacheve,Ukraine)
Publishing activity
The Department publishes two journals: the scientific journal “Theory of Proba-bility and Mathematical Statistics” and the popular scientific one “In the Worldof Mathematics”.
“Theory of Probability and Mathematical Statistics” was founded in 1970.Two issues appear annually. The editor-in-chief is A. Skorokhod, assistants to theeditor-in-chief are Yu. Kozachenko and M. Portenko, the editorial board secretary
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is A. Olenko. The journal is translated into English and published in the USA bythe American Mathematical Society.
“In the World of Mathematics” is the only scientific popular mathematics jour-nal in Ukraine. It is designed first of all for schoolchildren, but many materialsmay be useful and interesting also for students, teachers and everyone interestedin mathematics. The journal relates about the newest achievements in mathe-matics, unsolved problems and various applications of mathematics in differentspheres of knowledge. The goal of the journal is to develop the readers’ creativeabilities, involve them into scientific work, and also to search for gifted pupils.Therefore the journal gives a lot of information about mathematical olympiads,competitions, and creative contests. The members of the editorial board assistin organizing the Ukrainian mathematical Olympiads and selecting the team forinternational mathematical olympiads. The journal contains a lot of entertainingmaterials – mathematical games, mathematical fairy tales and detective stories,puzzles, rebuses. Interesting chess page and news of mathematical life are alsopresented in the journal.
Besides that, Department takes an active part in the edition of three otherscientific journals: “Theory of Stochastic Processes”, “Random Operators and Sto-chastic Equations” and “Applied Statistics, Actuarial and Financial Mathematics”.
Participation in TEMPUS-TACIS Projects
Training of high level professional specialists, competitive in the labor market,is impossible without the European teaching experience and collaboration withleading educational centers of Europe. Essential help to Ukraine in this area isbeing rendered by the European Union within the framework of the TEMPUS–TACIS Programme aimed at the development of cooperation between higher ed-ucation institutions of the European Union and Ukraine.
According to this programme the Department of Probability Theory andMathematical Statistics participated in the TEMPUS–TACIS JEP – 10353-97Project “Statistical Aspects of Economics” in cooperation with the Universities ofStockholm (Sweden), Umea(Sweden) and Helsinki (Finland) and in the TEMPUS–TACIS Network Project NP-22012 - 2001 “Improvement of Education in Statisti-cal Applications in Economics” in cooperation with the Universities of Stockholm(Sweden), Vasteras (Sweden) and Helsinki (Finland).
The creation of the Network of Ukrainian universities and practitioners“Economic-Statistical Education in Ukraine” was a key success of the NetworkProject. The founders of the Network were Kyiv National Taras ShevchenkoUniversity, Nizhyn Pedagogical University, Uzhgorod National University andUkrainian State Committee in Statistics. The great organizational work and suc-cessful dissemination actions attracted the non-consortium members and pro-moted increasing the number of network members: Lviv, Chernivtsi, Volyn(Lutsk), Donetsk, Dnepropetrovsk and Kherson Universities. All new memberswere involved in the Network activities: curricula/course development, exchangeof information, training and dissemination.
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Due to realization of the projects the Department of Probability Theory andMathematical Statistics of Kyiv University has become the center of implemen-tation and dissemination of know-how, modern teaching methodology and goodpractices in new economic-statistical specialities. These projects are good exam-ples of useful bilateral and multilateral dialogue and cooperation between Euro-pean and Ukrainian Universities.
Project IВ-JEР-25054-2004
Now Department of Probability Theory and Mathematical Statistics is partici-pating in Joint European Project IB-JEP-25054-2004 “Training Centre for Actuar-ies and Financial Analysts”, 2005–2008, which is carried out within the frameworkof TEMPUS Program (Trans-European Cooperation Scheme for Higher Educa-tion) financed by European Commission (Directorate-General for Education andCulture).
The Ukrainian participants of the project are: State Commission for Regu-lation of Financial Services Markets of Ukraine (Statefinservices), governmentalstructure responsible for qualification demands and certification of specialists el-igible to fulfil actuarial calculation according to the law “On Insurance”, KyivNational Taras Shevchenko University (academic institution which will developand deliver the training courses) and Society of Actuaries of Ukraine (SAU), non-state professional association.
The European participants of the project: Malardalen University (Sweden),Stockholm University (Sweden), Catholic University of Leuven (Belgium), Uni-versity of Cologne (Germany), University of Aegean (Greece).
The Project Coordinator is Professor Dmitrii Silvestrov (Malardalen Univer-sity, Vasteras, Sweden).
The strategic goal of this project is to assist in the development of actuarialprofession in Ukraine through the implementation of the system of actuaries train-ing. It presents several immediate steps straightforwardly responding demands ofinsurance/financial sector of economy. The project is aimed at establishing properprofessional standards and increasing the role of actuarial profession in Ukraineand therefore developing the strong insurance services industry in Ukraine as anessential element of social protection and civil society development.
13
Некоторые задачи, связанные с оценкойразорения страховой компании
Бондарев Б.В., Жмыхова Т.В., Украина
Пусть задан финансовый рынок, с возможностью инвестировать вбезрисковый актив B , цена которого эволюционирует согласно разностномууравнению:
∆Bn = rBn−1, B0 > 0 (1)Динамика капитала страховой компании, размещающей свой капитал u0 назаданном финансовом рынке, описывается уравнением:un+1 = un(1 + r) + c− Zn+1
где un - капитал компании в момент времени n ; c - размер страховых взносов;Zn+1 - выплаты по страховым искам, производимым компанией в моментвремени n+ 1 .
Вероятность разорения за бесконечное число шагов описываетсяуравнением:
ϕ(u0) = 1− Fz(u0(1 + r) + c) +u0(1+r)+cR
0
ϕ(u0(1 + r) + c− y)dFz(y)
которое получается из рекуррентного соотношения[1]:
ϕk+1(u0) = 1− Fz(u0(1 + r) + c) +u0(1+r)+cR
0
ϕk(u0(1 + r) + c− y)dFz(y)
здесь ϕ1(u0) = 1− Fz(u0(1 + r) + c) .Теорема. Пусть размеры исков имеют гамма-распределение с
параметрами n и α ,пусть у страховой компании имеется также возможностьв каждый момент времени размещать свой капитал на банковском депозите.Тогда вероятность разорения за бесконечное число шагов будет равна:
ϕ(x) =
∞Xk=0
dke−α(1+r)(Akx),
где Akx = (1 + r)kx+ c (1+r)k−1r
, k = 0, 1, ...
dk = d0(1−b)n(1−b2)n(1−b3)n...(1−bk)n , k = 1, 2, ... здесь b = 1 + r ,
∞Pk=0
dk = 1 .
Литература
[1] Мельников А.В. Риск-менеджмент:стохастический анализ рисков вфинансах и страховании.. М.: изд-во "Анкил", 2001. 112 с.
Кафедра теорiї ймовiрностей та математичної статистики,математичний факультет,Донецький нацiональний унiверситет,вул. Унiверситетська, 24, Донецьк 83055, Українаe-mail: [email protected]
14
О разностном аналоге задачи стабилизациистохастического интегрального уравнения
Вольтерра
Брадул Наталья Валерьевна, Украина
Рассмотрим стохастическое уравнение Вольтерра при постояннодействующих случайных возмущениях
x(t) = η(t) +
Z t
0
K(t, s)x(s)ds (1)
решение которого устойчиво в среднем квадратическом [1]. С помощью θ -метода построен разностный аналог уравнения (1)
x(i+1) = η(i+1)+θa(i, 0)x(0)+(1−θ)a(i, i)x(i)+
i−1Xj=1
a(i, j)x(j), 0 ≤ θ ≤ 1. (2)
Показано, что если решение исходного интегрального уравненияустойчиво, то существует θ и шаг дискретизации этого уравнения такие,что решение соответствующего разностного уравнения также устойчиво.Таким образом, получена область значений θ , при которых разностныйаналог (2) сохраняет свойство устойчивости исходного уравнения [2].Приведены примеры. Полученные результаты используются для решениязадачи стабилизации стохастического уравнения Вольтерра с квадратичнымфункционалом качества.
Литература
1. Шайхет Л.Е. Об устойчивости решений стохастических уравненийВольтерра.-АиТ, 1995.-с.93-102.
2. Брадул Н.В., Шайхет Л.Е. Задача оптимальной стабилизации длястохастического разностного уравнения Вольтерра.- Труды ИПММ, Т.9,2004.- с.24-45.
Донецкий государственный университет управления,83015, Донецк-15, ул. Челюскинцев 163а
e-mail: [email protected]
15
Про оцiнку ймовiрностi банкрутства длямоделi Крамера-Лундберга страховоїкомпанiї за можливостi iнвестування
капiталу в кiлька видiв акцiйБратик Михайло Васильович, Україна
Розглянемо модель процесу ризику страхової компанiї, капiтал якої вмомент часу t ≥ 0 описується рiвнянням
Y (t, x, φ1s, φ
2s) = x+ ct−
N(t)Pi=1
Xi +tR0
φ1sS1
sdS1
s +tR0
φ2sS2
sdS2
s
Перша компонента моделi R1(t) = x + ct −N(t)Pi=1
Xi є класичною: x ≥ 0 –
початковий капiтал; Xi – послiдовнiсть незалежних однаково розподiленихдодатних випадкових величин; N(t) – пуасcонiвський процес з iнтенсивнiстюλ , незалежний вiд Xi ; c > 0 – iнтенсивнiсть надходження премiй. Друга
компонента R2(t) =tR0
φ1sS1
sdS1
s +tR0
φ2sS2
sdS2
s вiдповiдає припущенню про те, що
страхова компанiя може iнвестувати весь свiй капiтал або його частину в двавиди акцiй, цiни яких Sit , i = 1, 2 є м.н. додатними i допускають стохастичнiдиференцiали вигляду
dSit = Sit(dMit + dAit), t ≥ 0, i = 1, 2,
де M it квадратично iнтегрованi неперервнi мартингали, квадратичнi
характеристики яких 〈M i〉t є абсолютно неперервнi вiдносно мiри Лебега,
〈M i〉t =tR0
βisds,tR0
|βis|ds <∞, t > 0, м.н.,
Ait – процеси iнтегрованої варiацiї, Ait =tR0
αisds,tR0
|αis|ds <∞, t > 0, м.н.
Процеси S1t та S2
t вважаються незалежними, а тому 〈M1,M2〉t ≡ 0 .Тут φit визначає кiлькiсть грошей, яку страхова компанiя в момент часу
t вкладає в i -тий вид акцiй, t ≥ 0 , i = 1, 2 .Будемо вважати, що процеси φit є F it -передбачуваними, де
F it = σR1(s), Sis,Mis, A
is, 0 ≤ s ≤ t та задовольняють умови
EtR0
(φis)2βisds < ∞ ,
tR0
|φisαis|ds < ∞ , i = 1, 2 . Припускаємо також, що
M1, M2 , A1 , A2 , φ1 , φ2 не залежать в сукупностi вiд Xi , i ≥ 1 та N .Розглянемо додатний випадковий процес
Z(t, x, φ1, φ2, r) = exp−rY (t, x, φ1, φ2) , r > 0
i позначимо
16
h(r) = E expX1 − 1 ,
f(t, φ1, φ2, r) = λh(r)− cr + r2
2((φ1
t )2β1t + (φ2
t )2β2t )− r(φ1
tα1t + φ2
tα2t ) .
Теорема 1 Нехай для деякого r > 0 i стратегiї φ1t , φ
2t , t ≥ 0 виконуються
умови
1) E expX1 <∞ ;
2) f(t, φ1, φ2, r) ≤ 0 , t ≥ 0 , м.н.
Тодi процес Z(t, x, φ1, φ2, r), t ≥ 0 – Ft -супермартингал.
Оскiльки ймовiрнiсть банкрутства страхової компанiї шукатимемо увиглядi ψ(x) ≤ e−rx , то параметр r доречно вибирати якнайбiльшим,зокрема бiльшим за єдиний додатний корiнь рiвняння λh(r) = cr ,бо оцiнку ймовiрностi банкрутства з цим коренем можна отримативзагалi не iнвестуючи в жодну з акцiй, тобто дотримуючись стратегiїφ1
s ≡ 0, φ2s ≡ 0, s ≥ 0 .
Теорема 2 Нехай функцiї αi та βi , i = 1, 2 задовольняють наступнiумови:
1) βis > 0 , s ≥ 0 , i = 1, 2 ;
2) iснують c1, c2 > 0 такi, що ess infs≥0((αi
s)2
2βis
) ≥ ci м.н., i = 1, 2 .
Тодi iснує ненульова передбачувана стратегiя φ1s, φ
2s, s ≥ 0 , що
задовольняє умову 2) теореми 1, i при цьому можна покласти r = r , де r –єдиний корiнь рiвняння λh(r) = cr + c1 + c2 . Для даної стратегiї має мiсцеоцiнка ймовiрностi банкрутства страхової компанiї ψ(x, φ1
s, φ2s) ≤ e−rx .
У випадку iнвестування в декiлька видiв акцiй, цiни яких є семiмартингалами,отримано оцiнку ймовiрностi банкрутства страхової компанiї ψ(x) ≤ e−rx , вякiй коефiцiєнт r покращено порiвняно з випадком вiдсутностi можливостiiнвестування чи при можливостi iнвестування капiталу тiльки в один видакцiй.
Лiтература
[1] Gaier J., Grandits P., Schachermayer W. Asymptotic ruin probabilities andoptimal investment. //The annals of applied probability, 2003, Vol.13, No.3,1054-1076.
[2] Мiшура Ю.С. Оцiнка ймовiрностей банкрутства для моделей здовгостроковою залежнiстю. //Теорiя ймовiрностей та математичнастатистика, 2005, 72, cт. 93-100.
Нацiональний унiверситет "Києво-Могилянська академiя",Київ, вул. Сковороди 2e-mail: [email protected]
17
Про зображення дискретних мартингалiв
В.В.Булдигiн, Г.В.Панфiлова, Україна
Нехай X1, X2, . . . , XN — незалежнi в сукупностi однаково розподiленiцентрованi випадковi величини, якi приймають l значень c1, c2, . . . , cl зiмовiрностями p1 > 0, . . . , pl > 0 , Fn = σ(X1, X2, . . . , XN ) , 1 ≤ n ≤ N ,F0 = ∅,Ω . Вiдомо, що у випадку l = 2 центрований мартингал(Mn, Fn, 1 ≤ n ≤ N) можна розкласти за “базисною” послiдовнiстю (Xn) :
Mn =
nXk=1
αkXk, 1 ≤ n ≤ N, (1)
де випадковi величини αk є Fk−1 -вимiрними, k ≥ 1 .Разом з зображенням (1) ми розглядаємо бiльш загальне зображення:
Mn =
nXk=1
αkβk, 1 ≤ n ≤ N, (2)
де випадковi величини αk є Fk−1 -вимiрнi, k ≥ 1 , та βk є σ(Xk) -вимiрнi,k ≥ 1 .
В роботi знаходяться необхiднi i достатнi умови, за яких при l ≥ 2мартингал (Mn, Fn, 1 ≤ n ≤ N) можна зобразити у виглядi (2). Це даєможливiсть побудувати приклад, який показує, що при l > 2 зображення(2), i разом з тим зображення (1), не має мiсця.
Нацiональний технiчний унiверситет України “КПI”, м. Київ
18
Про асимптотичну поведiнку розв’язкiвдеяких стохастичних диференцiальних
рiвнянь
В.В.Булдигiн, О.А.Тимошенко, Україна
Нехай додатнi неперервнi функцiї g(·) та σ(·) такi, що стохастичнедиференцiальне рiвняння dX(t) = g(X(t)) dt+σ(X(t)) dW (t) , t ≥ 0 , X(0) ≡ 1 ,має м.н. єдиний неперервний розв’язок X(·) та limt→∞X(t) = ∞ м.н.Розглянемо питання про асимптотичну стiйкiсть розв’язкiв цього рiвняннявiдносно функцiї g(·) . А саме, якими повиннi бути функцiї g1(·) та g2(·) , щобX1(·) ∼ X2(·) , тобто limt→∞
X1(t)X2(t)
= 1 м.н. Припущення, що gk(·) , k = 1, 2 ,повиннi бути такими, щоб g1(·) ∼ g2(·) , не є достатнiм для еквiвалентностiX1(·) та X2(·) . Слiдуючи [1], розглянемо деякi додатковi умови, за яких iзg1(·) ∼ g2(·) випливає X1(·) ∼ X2(·) м.н.
Покладемо Gk(t) =R t1
dugk(u)
, t ≥ 1 , k = 1, 2 .
Теорема 1 Нехай
limt→∞
G1(t) =
Z ∞
1
du
g1(u)= ∞, (1)
lim inft→∞
Z ct
t
du
g1(u)G1(u)> 0 для всiх c > 1. (2)
Тодiа) якщо G1(·) ∼ G2(·) , то X1(·) ∼ X2(·) м.н.,б) якщо limc↓1 lim supt→∞
R ctt
dug1(u)G1(u)
= 0 , то X1(·) ∼ X2(·) м.н. тодi iтiльки тодi, коли G1(·) ∼ G2(·) .
Теорема 2 Нехай виконуються умови (1) та (2). Тодi, якщо g1(·) ∼ g2(·) ,то X1(·) ∼ X2(·) м.н.
Лiтература
[1] Булдигiн В.В., Клесов О.I., Штайнебах Й.Г., PRV властивiсть функцiй таасимптотична поведiнка розв’язкiв стохастичних диференцiальних рiвнянь.— Теор. ймовiрн. та мат. стат., Вип. 72, 2004, стор. 63–78.
Нацiональний технiчний унiверситет України “КПI”, м. Київ
19
Анализ связей между результатамиобработки смесей и характеристиками
действовавших на них случайных импульсов
Васильев Сергей Николаевич, Украина
В докладе исследуются связи между динамикой процентного содержаниякомпонент смеси и возмущениями, которые эти изменения порождают.
Изучен процесс формирования компонент жидкой смеси при воздействиина неё импульсов, локализованных в малых частях занимаемого ею объёма.Высвобождающаяся при этом энергия приводит к изменениям свойств смесив частях объёма, подвергшихся возмущениям. Многократные возмущениявсех частей объёма, приводят к изменению свойств смеси во всём объёме.Предполагается, что: характеристики импульсов и частота их реализации(во времени) являются случайными величинами; характеристики импульсовв различных частях объёма, вообще говоря, не совпадают.
Установлены условия, при выполнении которых процесс формированиякомпонент смеси будет стабилизирован. Стабилизацией процесса обработкисмеси называется такое её состояние, при котором последующие возмущениясмеси не приводят к существенным изменениям её свойств.
Изменение характеристик многокомпонентной смеси достигается засчёт воздействия на неё быстро изменяющихся во времени импульсов.После каждой серии таких возмущений устанавливается - какие измененияпретерпели характеристики смеси.
Если обработка исходной смеси производилась с помощью возмущений,тождественных по своим характеристикам и возмущения были равномернораспределены во времени и в занимаемой ею объёме V , то полученнаяв результате смесь будет однородной в V . Если же в заданном объёмеформируется смесь, свойства которой в разных её частях в конце обработкидолжны иметь разные характеристики, то и характеристики возмущенийв различные моменты времени, в различных частях V , должны бытьнеодинаковы.
Установлено, что при определённых способах воздействия на смесьпроцесс получения её компонент может быть стабилизирован: процентноесодержание компонент в обрабатываемой смеси в малой окрестности любойточки из V будет при дальнейших ее возмущениях либо сохраняться, либомедленно изменяться в заданных границах.
ГП Харьковский приборостроительный завод им. Т.Г. Шевченко,Харьков, 61004, ул. Октябрьской революции, 99e-mail: [email protected], [email protected]
20
Iнтерполяцiйнi зображення одного класувипадкових полiв
Верьовкiна Ганна Володимирiвна, УкраїнаНагорний Володимир Никифорович, Україна
Нехай ξ(t) , t ∈ R2 - сепарабельне випадкове поле [1] з Mξ(t) = 0 , яке маєзображення виду
ξ(t) =
ZΛ×Λ
2Yi=1
fi(ti, λi)Z(dλ), (1)
де Λ - деяка множини параметрiв, Z(dλ) - випадкова функцiя множин наΛ× Λ така, що
MZ(dλ) = 0;MZ(A1, A2)Z(B1, B2) = F (A1, A2, B1, B2), (2)де F - комплексна функцiя множин, адитивна за всiма аргументами, додатновизначена i така, що Z
Λ2×Λ2|F (dλ, dµ)| < +∞. (3)
Нехай функцiї fi(ti, λi) , i = 1, 2 такi, що кожну з них можна довизначитив комплекснiй площинi вiдносно ti до цiлої функцiї експоненцiального типуз показником ci(λi) таким чином, що
supλi∈Λ
ci(λi) = σi <∞; supλi∈Λ
supti∈R
|fi(ti, λi)| = Li <∞. (4)
Має мiсце наступна теорема.
Теорема 1 Нехай ξ(t) - сепарабельне випадкове поле виду (1), для якоговиконанi умови (2)-(4). Тодi з ймовiрнiстю одиниця має мiсце зображення
ξ(t) =
∞Xk1=−∞
∞Xk2=−∞
ξk1,k2(t)ωk1,k2(t) (5)
У формулi (5) ξk1,k2(t) та ωk1,k2(t) визначаються певним чином.
Лiтература
[1] Ядренко М.Й. Аналiтичнi випадковi поля // Вiсник Київськогоунiверситету.-Серiя математика та механiка,-Вип.11,-1969
Київський нацiональний унiверситет iменi Тараса Шевченка,механiко-математичний факультетe-mail: [email protected]
21
Случайные блуждания на множествеграфов с изменяющимся числом вершин
Вовк Александр Владимирович, Дикарев Вадим Анатольевич, Украина
Рассмотрена задача о стабилизации процессов случайных блужданий намножестве графов, имеющих непустое пересечение (центр). Общим центромбудем называть такой граф Γ0 ∈ Γi который имеет непустые пересечениясо всеми графами из Γi : Γ0
TΓi 6= ∅, i = 0, 1, ..., n . Установлено,
что для любой заданной эволюции центра можно подобрать возмущениявероятностных связей между вершинами графов, не входящих в центр, так,чтобы эволюция процесса случайных блужданий на множестве графов былареализована за любой промежуток времени, с любой заданной точностью.Условия, которым должны удовлетворять распределения, действующие намножестве графов Γi с общим центром, состоят в следующем:
1. Взаимное положение графов не изменяется с изменением времени.
2. После каждого возмущения Γi вектор распределения pi(t) процесса нанем не изменяется до следующего возмущения Γi или его части.
3. Условия согласования. Пусть ΓiT
Γi = Γij 6= ∅ , pij - векторраспределения процесса случайных блужданий на Γij . Обозначимчерез pij , pji подвекторы векторов распределений pi , pj на Γij .Считаем что векторы pij , pji коллинеарны.
Доказано, что при многократных возмущениях всех графов, выбранныхспециальным образом, вероятности состояний процесса либо принимаютпредельные значения, либо локализуются вблизи них. Основными условиями,которые приводят к стабилизации, являются быстро изменяющиесяво времени факторы, вызывающие сильные возмущения основныххарактеристик процесса.
Задача о стабилизации процесса случайных блужданий на множествеграфов с общим центром была исследована и для случая, когда числовершин в нём изменяется с изменением времени. Чаще всего изменениечисла вершин приводит к дестабилизации процесса случайных блужданий.Получены условия, при выполнении которых изменение числа вершин неприводит к его дестабилизации.
Полученные результаты дают возможность исследовать процессы вотдельных фрагментах сети Internet и производить анализ работы следящихсетей. Такие сети используют, в частности, при исследовании процессов,происходящих в активных смесях.
Харьковский национальный университет радиоэлектроники,Харьков, 61166, пр. Ленина,14e-mail: rurouni_v@ mail.ru
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Стабилизация распределений марковскихцепей с переменным числом состояний за
конечное времяГерасин С.Н., Украина
В докладе будут рассмотрены марковские цепи с изменяющимсячислом состояний и способы стабилизации их распределений. Можновыделить классы распределений (назовем их стационарными), которые дляописываемых цепей могли бы выполнять роль стационарного распределения(в том смысле, что если начальное распределение цепи принадлежит данномуклассу, то и во все последующие моменты времени распределения цепи будутпринадлежать тому же классу).
Будем рассматривать цепи, в которых последовательные множествавозможных состояний системы I1, I2, . . . , Ik, Ik+1, . . . отличаются другот друга на одно состояние. Переходная матрица Pk , определяющаяпереход с увеличением числа состояний, может быть представлена в видеPk = [B
(n)k , Ck] , где B
(n)k - квадратная матрица порядка n (размерность
Pk − n × n + 1), полученная из Pk вычеркиванием последнего столбца
Ck . Такие матрицы могут быть представлены в виде Pk =
„U
(n−1)k
rk
«,
где U(n−1)k = P
(n−1)k - квадратная стохастическая матрица порядка
n − 1 , а rk - строка с неотрицательными элементами такая, чтоrkj ≥ 0(1 ≤ j ≤ n− 1),
Pn−1j=1 rkj = 1 .
Рассмотрим условия сходимости распределения данной цепик распределению из стационарного класса. Всякое распределениерассматриваемой цепи p(k) в момент времени k может быть представлено ввиде p(k) = πk + f(k) , где πk ∈ π является левым единичным собственнымвектором либо для P
(n)k (U
(n−1)k ) , либо для P
(n+1)k+1 , f(k) - вектор-строка с
нулевой суммой компонент, так называемый ”возмущающий” вектор.Для того чтобы сходимость к стационарному классу имела место,
достаточно, чтобы в f(k) стремился к вектору, все компоненты которогоравны нулю. Это имеет место, в частности, в том случае, когдаразмерность подпространства возможных значений возмущающего вектораf(k) (”покрывающего” подпространства) с каждым переходом уменьшается.Рассмотрим условия, при которых такое уменьшение размерности имеетместо.
Положим, что матрицы P(n)k и U
(n−1)k не имеют других собственных
чисел, кроме нуля и единицы, причем последнее имеет кратность1 . Однородные цепи, порождаемые такими матрицами, достигаютстационарного распределения за конечное время, не превышающееразмерности наибольшей нулевой клетки в жордановой нормальной формесоответствующей матрицы, что сопровождается уменьшением размерностипокрывающего подпространства. В случае неоднородной цепи с переменнымчислом состояний это условие, для получения аналогичных результатов,
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необходимо дополнить некоторыми ограничениями в плане ”согласованности”жордановых нормальных форм и обобщенных собственных векторов,следующих друг за другом матриц P
(n)k и U
(n−1)k (разной размерности).
Обозначим mk - число нулевых клеток в жордановой нормальной формематрицы P
(n)k или U
(n−1)k ; vki - размерность i -й (1 ≤ i ≤ mk ) нулевой
клетки в указанной форме; fij(k) - левый присоединенный вектор j − 1 -го порядка (собственный при j = 1) в i -й группе левых обобщенныхсобственных векторов собственного значения 0 матрицы P
(n)k или U
(n−1)k
(1 ≤ i ≤ mk, 1 ≤ j ≤ vki ) .
Лемма 1 Для того, чтобы при последовательном увеличении числасостояний во время переходов в цепи происходило уменьшение размерностиподпространства, ”покрывающего” возможные значения возмущающеговектора f(k) , достаточно, чтобы:
1) жордановы нормальные формы матриц P(n)k и P
(n+1)k+1 отличались
только либо ”появлением” ”новой” нулевой клетки размерности 1 , либо”увеличением” размерности одной из ”старых” нулевых клеток на единицу;
2) n+1 -я (”новая”) компонента отлична от нуля лишь либо у собственноговектора, отвечающего ”новой” нулевой клетке, либо у присоединенноговектора наивысшего порядка в группе, ”увеличившей” свой состав;
3) остальные нулевые обобщенные собственные вектора матрицы P(n+1)k+1
связаны с соответствующими векторами матрицы P(n)k соотношениями:
fij(k + 1) = (fij(k + 1), 0), 1 ≤ i ≤ mk, 1 ≤ j ≤ vki ,
fij(k) =Pmkg=1
P(j∧vkg )
l=1 βijglfgl(k + 1) , где a ∧ b = min(a, b) .
Заметим, что всякий вектор с нулевой суммой компонент, подаваемыйна вход матрицы, описывающей такой переход, может быть представлен в
виде„−cπk+
Pmki=1
P(vki )
j=1 αijfij(k), c
«, где πk - левый единичный собственный
вектор матрицы U(n−1)k ; mk , vki , fij(k) имеют тот же смысл, что и ранее; c
скаляр, равный ”исключаемой” компоненте вектора, подаваемого на вход.
Теорема 1 Пусть начиная с некоторого момента k0 , на протяжении, покрайней мере, N переходов (N = min(dim(Pk0 , P
′k0),dim(P ′, Pk0)) , где A′ -
матрица, транспонированная по отношению к матрице A) выполняютсяусловия одной из лемм 1. Тогда к моменту k0 + N распределениевероятностей состояний цепи будет принадлежать стационарномусогласованному классу данной цепи вне зависимости от начальногораспределения.
Харьковский национальный университет радиоэлектроники,кафедра высшей математики,Харьков 166, пр. Ленина, 14e-mail: [email protected]
24
Умови належностi розподiлу випадковоївеличини з незалежними s-адичними
цифрами до класу мiр РайхманаЯ.В.Гончаренко, I.О.Микитюк, Україна
Нагадаємо, що характеристичною функцiєю випадкової величиниζ називається комплекснозначна функцiя fζ(t) = Meitζ , де M —математичне сподiвання. Вiдомо, що коли ζ має дискретний розподiл, тоLζ = lim
|t|→∞sup |fζ(t)| = 1. Якщо розподiл ζ є абсолютно неперервним, то
Lζ = 0 ; якщо — сингулярним, то 0 ≤ Lζ ≤ 1 . Отже, за поведiнкою модуляхарактеристичної функцiї на нескiнченностi (тобто за величиною Lζ ) можначастково судити про тип розподiлу. У випадку, коли Lζ = 0 , то мiра, щовiдповiдає розподiлу ζ , називається мiрою Райхмана.
Нехай s — фiксоване натуральне число, бiльше 1 . Розглядається
випадкова величина ξ =∞Pk=1
s−kηk, s -адичнi цифри ηk якої є незалежними,
причому ηk набувають значень 0 , 1 , ..., s − 1 з ймовiрностями p0k , p1k , ...,p(s−1)k вiдповiдно (pak ≥ 0 , p0k + p1k + ...+ p(s−1)k = 1). Випадкова величинаξ задовольняє умови теореми Джессена-Вiнтнера i тому має чистий розподiл.Необхiднi i достатнi умови належностi розподiлу ξ до кожного з чистих типiввiдомi [1]. Разом з цим, ми не виявили дослiджень, в яких би розв’язуваласьзадача про поведiнку модуля характеристичної функцiї випадкової величиниξ на нескiнченностi.
В доповiдi пропонуються необхiднi i достатнi умови для Lξ = 0 .
Лемма 1 Для довiльного додатного t добуток
Dt =
∞Yk=w(t)+1
|fk(t)|, де
w(t) = 2 +
»logs
t
2π
–, (1)
збiгається, причому Dt ≥ C > 0 , де
C =1√s− 1
∞Yj=2
„1− π2
2s2j
«. (2)
Наслiдок 1 Для модуля характеристичної функцiї fξ(t) випадковоївеличини ξ мають мiсце нерiвностi
At · C ≤ |fξ(t)| ≤ At ∀t ∈ R+,
де At =w(t)Qk=1
|fk(t)| , а константа C визначається рiвнiстю (2), а число w(t)
— рiвнiстю (1).
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Наслiдок 2 Для модуля характеристичної функцiї fξ(t) випадковоївеличини ξ мають мiсце спiввiдношення
|fξ(t)| → 0 (t→∞) ⇔ At → 0 (t→∞).
Теорема 1 Для характеристичної функцiї fξ(t) випадкової величини ξ маємiсце рiвнiсть |fξ(t)| = 0 тодi i тiльки тодi, коли iснує k таке, що|fk(t)| = 0
Теорема 2 Якщо для характеристичної функцiї fξ випадкової величини ξмає мiсце рiвнiсть Lξ = 0 , то
limn→∞
Mn = 0, (3)
де Mn = |fn+1(2snπ)| · |fn+2(2snπ)|,
|fn+j(2πsn)|2 =
„s−1Pa=0
pa(w+j) cos 2πs−j«2
+
„s−1Pa=0
pa(w+j) sin 2πs−i«2
=
= 1− 4s−1Pr=1
„sin2 rπ
sj
s−r−1Pa=0
p(a+r)(n+j)pa(n+j)
«, j = 1, 2.
(4)
Теорема 3 Якщо ∀h = 1, 2, ...,ˆs2
˜|fn+1(2πsnh)| → 0 (n→ 0) або |fn+2(2πsnh)| → 0 (n→∞),
то Lξ = 0.
Лiтература
[1] Albeverio S., Gontcharenko Ya., Pratsiovytyi M., Torbin G. Convolutions ofdistributions of random variables with independent binary digits. — PreprintSFB-611, Bonn, 2002. — No 23. — 20 p.
[2] Працьовитий М.В. Фрактальний пiдхiд у дослiдженнях сингулярнихрозподiлiв. — Київ: Вид-во НПУ iменi М.П.Драгоманова, 1998. — 296 с.
[3] Працьовитий М.В. Згортки сингулярних розподiлiв // Доп. НАНУкраїни. — 1997. — No 9. — C.36–42.
[4] Турбин А.Ф., Працевитый М.В. Фрактальные множества, функции,распределения. — Киев: Наукова думка, 1992. — 208с.
[5] Гончаренко Я.В. Асимптотичнi властивостi характеристичної функцiївипадкової величини незалежними двiйковими цифрами та згорткисингулряних розподiлiв ймовiрностей // Науковi записки НПУ iменiМ.П.Драгоманова. Фiзико-математичнi науки. — 2002. — No 3. — С. 376-390.
НПУ iменi М.П.Драгоманова, Київ, УкраїнаВолинський державний унiверситет, Луцьк, Україна
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Использование следящих графов приисследовании случайных полей,возникающих в активных средах
Гора Николай Николаевич, Украина
Формирование жидких многокомпонентных смесей обычно производитсяв специальных устройствах - муфтах. Ядро муфты представляет собойрезервуар, заполненный обрабатываемой смесью. В него, по специальнымканалам (трубам), поступают реагенты - активные компоненты, с помощьюкоторых производится управление процессом. На практике слежениеза процессом формирования жидкой смеси связано с определённымитрудностями. Одна из них состоит в том, что фиксация основныххарактеристик смеси не всегда может быть произведена, поскольку доступв отдельные части ядра муфты, затруднителен. Кроме того, протекающиев муфте реакции подвергаются массе случайных воздействий. Поэтомуисследуемый процесс является случайным. В этом докладе предложен новыйспособ определения основных характеристик процесса формировании смеси.Он состоит в следующем: в ядре муфты и в частях труб, примыкающихк ядру, располагается следящая сеть, позволяющая фиксировать данныео квазистатическом поле в муфте. Следящая сеть представляет собойграф, в вершинах которого располагаются датчики, с помощью которыхфиксируются потенциалы квазистатического поля, возникающего в муфтеиз-за происходящих в ней химических реакций. О величине потенциалаэтого поля в точках расположения датчиков можно судить по частотеразрядов происходящих в них. Если во всех датчиках эти частоты лежатв заданных границах, то процесс протекает нормально. В противном случае,в отдельных частях муфты, где частота "проскоков" разряда не попадает взаданные границы, процесс отклоняется от заданного режима. Потенциалыэлектростатических полей, возникающих в обрабатываемой смеси, являютсяважной характеристикой, протекающих в ней химических процессов. Попотенциалам этих полей можно судить о том, протекает ли процессформирования смеси в заданных нормативных границах или нет. Данныерезультатов наблюдений, в которых были зафиксированы характеристикиквазистатического поля в различных точках объёма, занимаемого смесью,были подвергнуты статистической обработке. Полученные результаты даютвозможность исследовать динамику изменений характеристик поля, атакже те изменения его характеристик, которые происходят под действиемвозмущений.
ГП Харьковский приборостроительный завод им. Т.Г. Шевченко,Харьков, 61004, ул. Октябрьской революции, 99e-mail: [email protected], [email protected]
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Використання f -розкладiв дляпредставлення дiйсних чисел
Iванна Григор’єва, Україна
Пiд представленням чисел розумiють подання чисел за певним правиломза допомогою цифр з визначеного алфавiту (наприклад, десятковепредставлення, двiйкове). В останнiй час уявлення про рiзнi способипредставлення дiйсних чисел значно розширились. Сьогоднi у математицiпоряд з традицiйними широко використовуються нетрадицiйнi способиподання чисел.
Поняття представлення дiйсних чисел з використанням f -розкладiввиникло з намагань узагальнити метричну теорiю ланцюгових дробiв.
Серед учених, якi працювали у напрямку вивчення f -розкладiв можнаназвати наступних: B. H. Bissinger [1], C. I. Everett [2], A. Renyi [3], F. Schweiger[4].
Розглянемо поняття f -розкладу числа.Нехай f — деяка фiксована додатна, монотонна функцiя, x — фiксоване
дiйсне число. Пiд f -розкладом числа x розумiють його представлення увиглядi
x = a0 + f(a1 + f(a2 + . . . ) . . . ), (1)
де цифри ai(x) та остачi ri(x) визначаються рекурсивно:
a0(x) = [x], r0(x) = x, (2)
an+1(x) = [ϕ(rn(x))], rn+1(x) = ϕ(rn(x)), n = 0, 1, 2, . . . , (3)
функцiя x = ϕ(y) є оберненою до y = f(x) .Якщо функцiя f спадна, то вiдповiдний f -розклад називають f -
розкладом типу А. Якщо ж f — зростаюча функцiя, то f -розклад називаютьf -розкладом типу Б.
Вiдомi достатнi умови iснування для дiйсного числа x його f -розкладу.
Теорема 1 Якщо для функцiї f виконуються умови
1. f(1) = 1 ;
2. f(x) — додатна, неперервна i строго спадна для 1 ≤ x ≤ G ,f(G) := lim
x→G− 0f(x) = 0 , 2 < G ≤ ∞ ;
3. |f(x2)− f(x1)| ≤ |x2 − x1| , 1 ≤ x1 < x2 ;∃λ : 0 < λ < 1
|f(x2)− f(x1)| ≤ λ|x2 − x1| , 1 + f(2) < x1 < x2 ,
то довiльне дiйсне x може бути представлене у виглядi f -розкладу типуА, який збiгається до цього числа x .
28
Теорема 2 Якщо для функцiї f виконуються умови
1. f(0) = 0 ;
2. f(x) — неперервна, строго зростаюча функцiя для 0 ≤ x ≤ G ,f(G) = lim
x→G− 0f(x) = 1 , 1 < G ≤ ∞ ;
3.f(x2)− f(x1)
x2 − x1< 1 , 0 ≤ x1 < x2 ,
то довiльне дiйсне x може бути представлене у виглядi f -розкладу типуБ, який збiгається до цього числа x .
Доведення цих фактiв мiститься у A. Reniy [3].Досить простим прикладом представлення дiйсних чисел за допомогою
f -розкладу типу А може виступати ланцюгове представлення дiйсних чисел.У цьому випадку f(x) = 1/x .
Як приклад представлення дiйсних чисел з використанням f - розкладутипу Б можна розглядати s - адичне представлення дiйсного числа x
x =
∞Xj=0
ajs−j , s ≥ 2, s ∈ Z . (4)
У цьому випадку f(x) = x/s .Дослiджуються також iншi властивостi f -розкладiв. Цiкавими є питання
єдиностi та скiнченностi (нескiнченностi) f -розкладу. Розглядаютьсяприклади.
Лiтература
[1] Bissinger B.H. A generalization of continued fractions.// Bull. Amer. Math.Soc., —1944. —50 — P. 868–876
[2] Everett C.I. Representations for real numbers.// Bull. Amer. Math. Soc., —1946. —52 — P. 861–869
[3] Renyi A. Representations for real numbers and their ergodic properties. //Acta Math. Acad. Sci. Hungar. — 1957. — 8 — P. 477–493
[4] Schweiger F. Ergodic theory of fibred systems and metric number theory. //Clarendon Press. Oxford. —1995. —P. 43-46.
Нацiональний педагогiчний унiверситет iменi М.П. Драгомановаe-mail: [email protected]
29
Застосування методу Фур’є длягiперболiчного рiвняння з ϕ-субгауссовою
правою частиною
Богдан Довгай, Україна
Розглянемо задачу
∂
∂x
„p(x)
∂u
∂x
«− q(x)u− p(x)
∂2u
∂t2= −p(x)ξ(x, t), x ∈ [0, π], t ∈ [0, T ] (1)
u|t=0 = 0,∂u
∂t
˛t=0
= 0, x ∈ [0, π] (2)
u|x=0 = 0, u|x=π = 0, t ∈ [0, T ], (3)
де T > 0 — деяка стала, функцiя p(x) —тричi неперервно диференцiйовна,p(x) > 0 , q(x) – неперервно диференцiйовна, q(x) ≥ 0 , ξ(x, t) – вибiрковонеперервне з iмовiрнiстю 1 строго ϕ -субгауссове випадкове поле, ϕ(x) = |x|pпри |x| > 1, p > 1 .
Нехай коварiацiйна функцiя B(x, y, t, s) випадкового поля ξ(x, t)задовольняє умовам:
1) У продовженої на всю площину по x, y функцiї B(x, y, t, s) iснуютьнеперервнi похiднi
∂i+jB(x, y, t, s)
∂xi∂yj, 0 ≤ i+ j ≤ 4;
2) Для B∗(x, y, t, s) = ∂2
∂x∂y
“p(x)p(y) ∂2
∂x∂yB(x, y, t, s)
”для довiльного
β ∈ [0, π] виконується умова:
supy∈[0,π]t∈[0,T ]s∈[0,T ]
Z π
−π
˛˛∆β
B∗(x, y, t, s) · 1p
p(x)
!˛˛ dx ≤ C
| lnβ|2γ
для деяких C > 0, γ > 2− 1p, де ∆βf(x, y, t, s) = f(x+β, y, t, s)−f(x, y, t, s) .
3) Iснують такi C′ > 0, δ > 1 − 1p
та β > 0 , що для довiльних t, s ∈ [0, T ]таких, що |t− s| < β мають мiсце нерiвностi
supx∈[−π,π]y∈[−π,π]
˛∂i+j
∂xi∂yjeB(x, y, t, s)
˛≤ C′
˛ln |t− s|
˛−2δ, 0 ≤ i ≤ 2, 0 ≤ j ≤ 2,
де eB(x, y, t, s) = B(x, y, t, t)−B(x, y, t, s)−B(x, y, s, t) +B(x, y, s, s) .
30
Тодi з iмовiрнiстю 1 iснує двiчi неперервно диференцiйовний розв’язокзадачi (1)–(3) в областi 0 ≤ x ≤ π, 0 ≤ t ≤ T , який можна зобразити увиглядi збiжного за ймовiрнiстю в нормi C ([0, π]× [0, T ]) ряду
u(x, t) =
∞Xn=1
Xn(x)1
µn
Z t
0
sinµn(t− u)ζn(u) du,
де
ζn(t) =
Z π
0
ξ(x, t)Xn(x)p(x) dx,
Xn(x) – ортонормованi з вагою p власнi функцiї, а µ2n – вiдповiднi власнi
значення задачi Штурма-Лiувiлля
d
dx
„pdX
dx
«− qX + λpX = 0
X(0) = X(π) = 0.
Лiтература
[1] Dovgay B.V., Kozachenko Yu.V. The conditions for application of Fouriemethod to the solution of nonhomogeneous string oscillation equation withϕ -subgaussian right side // Random operators and stochastic equations, 2005,Vol. 13, No. 3, pp. 281–296
[2] Довгай Б.В. Властивостi розв’язку неоднорiдного гiперболiчного рiвнянняз випадковою правою частиною // Український математичний журнал,2005, т. 57, 4, сс. 474–482
[3] Булдыгин В.В., Козаченко Ю.В. К вопросу применимости метода Фурьедля решения задач со случайными краевыми условиями // Случайныепроцессы в задачах математической физики. Сборник научных трудов. –Киев, Институт математики АН УССР, 1979. – С. 4 – 35.
[4] Булдыгин В.В., Козаченко Ю.В. Метрические характеристики случайныхвеличин и прецессов. – Киев: ТВiМС, 1998. – 289 с.
Київський нацiональний унiверситет iменi Тараса Шевченка,факультет кiбернетики,кафедра дослiдження операцiйe-mail: [email protected]
31
Застосування перiодограмних оцiнок доаналiзу автомодельних часових рядiв
Олексiй Зражевський, Україна
Основною темою роботи є застосування перiодограмних оцiнок до аналiзуавтомодельних часових рядiв з сильною залежнiстю, для яких параметрХюрста приймає граничнi значення близькi до 0.5 та 1.
Останнiм часом з’явилось багато дослiджень автомодельних часовихрядiв рiзних типiв та походжень. Наприклад, однiєю з актуальних проблемстатистики є дослiдження автомодельностi iнтенсивностi обмiну черезмережу Iтернет. [5]
Одним з методiв дослiдження автомодельностi та сильної залежностiчасового ряду є визначення параметра Хюрста. У лiтературi наведенобагато рiзних статистичних методiв аналiзу часового ряду з точки зоруавтомодельностi i визначення параметра Хюрста. Найбiльш вiдомими є методвибiркової дисперсiї агрегованого ряду, метод нормованого розмаху, метод,що базується на побудовi автокореляцiйної функцiї, метод перiодограм,метод Робiнсона (див. [1], [5]). Останнi три методи базуються на побудовiоцiнки спектральної щiльностi (i пов’язанною з нею перетворенням Фур’єавтокореляцiї). Показано (див. [2]), що за деяких умов перiодограмане є єффективною оцiнкою для спектральної щiльностi i не можезастосовуватись, наприклад, до аналiзу данних, близьких до бiлого шуму. Вцих випадках замiсть перiодограми використовуються перiодограмнi оцiнкизi спектральними вiкнами. Прикладами спектральних вiкон є вiкно Бартлета,Парзена, Т’юкi (див. [3], [4]).
В роботi розглянуто часовi послiдовностi для MSFT (тикер акцiй фiрмиMicrosoft) даних по цiнам акцiй i доходностi. Данi, що використовуються,були зареєстрованi за перiод з 02.01.2004 по 30.01.2004, в часовому промiжку09:30:00 - 16:00:00. Цiни тiкера приведенi до нульового середнього таодиничного середньоквадратичного вiдхилення. Для цих часових рядiвпроведенi дослiдження автомодельнiстi i побудованi оцiнки параметраХюрста.
Для визначення оцiнки параметра Хюрста для цiн на тiкер MSFT булизастосованi метод нормованого розмаху, метод побудови автокореляцiйноїфункцiї, метод перiодограм, метод Робiнсона. Вiдповiднi результати: 0.993,0.953, 0.962, 0.977. Перiодограмна оцiнка була покращена за допомогою вiкнаБартлета, вiкна Хеммiнга (вiкно Т’юкi з параметром 0.23) та вiкна Т’юкi зпараметром 0.28. Вiдповiднi результати: 0.966, 0.999, 0.998.
Для визначення оцiнки параметра Хюрста для доходностi тiкера MSFTбули застосованi метод вибiркової дисперсiї агрегованого ряду й методнормованого розмаху з вiдповiдними результатами: 0.512, 0.513. Отже можна
32
зробити висновок, що даний часовий ряд близький до бiлого шуму iперiодограма, як оцiнка спектральної щiльностi, не може застосовуватисьв цьому випадку. Для подальшого аналiзу були застосованi перiодограмнiоцiнки з вiкнами Бартлета i Т’юкi (з параметрами 0.23 та 0.28) зприпущенням, що спектральна щiльнiсть даного часового ряду є близькоюдо асимтотичної константи. Вiдповiднi результати: 0.505, 0.503, 0.504.
На основi проведених в роботi дослiджень можна зробити висновок,що часовi ряди цiн тiкера MSFT та доходностi тiкера MSFT пiдкоряютьсясильнiй залежностi, тобто є автомодельними. Часовий ряд цiн тiкера MSFT єпередбачуваним i для його аналiзу на автомодельнiсть можна застосовувативсi наведенi методи, причому перiодограмна оцiнка може бути покращеназа допомогою спектральних вiкон. Часовий ряд доходностi тiкера MSFTблизький до бiлого шуму, тобто є некорельованим i для його аналiзу необхiдновикористовувати згладженi за допомогою вiкон перiодограмнi оцiнки.
Статистичнi методи, якi використовувались в роботi, призначенi дляаналiзу асимптотичної поведiнки часових рядiв i тому виникають труднощi звикористанням їх для скiнченних вибiрок. Удосконалення вiдомих методiв iствореннях нових є однiєю з проблем, яка iснує на сьогоднiшнiй день i стоїтьперед фахiвцями з аналiзу часових рядiв.
Лiтература
[1] M. Moklyachuk, A. Zrazhevsky, Long-range dependence of time series forMSFT data of shares and returns, Vol.14, 4, 2006.
[2] А. Н. Ширяев, Вероятность - М.:Наука, 1989.-640 с.
[3] И. Г. Журбенко, И. А. Кожевникова, Стохастическое моделированиепроцессов - М.:Изд-во МГУ, 1990.- 148 с.
[4] S. Marple Lawrence, Jr., Digital Spectral Analysis with applications - Prentice-Hall Inc., Englewood Cliffs, 1987.- 848 с.
[5] M.S. Taqqu, W. Willinger, R. Sherman and D.V. Wilson, Self-SimilarityThrough High-Variability: Statistical Analysis of Ethernet LAN Traffic at theSource Level - IEEE/ACM Trans. Network., V.5, p.71-86, 1997.
Механiко-математичний факультет,Київський нацiональнiй унiверситет iменi Тараса Шевченка,вул. Володимирська, 64, 01033 Київ, Українаe-mail: [email protected]
33
Про операторне перетворення мартингала,пов’язаного з гiллястим випадковим
блуканнямОлександр Iксанов, Україна
Негадайлов Павло, Україна
Нехай S(·) - точковий процес на прямiй з точками A1, A2, . . . . Будеморозглядати гiллясте випадкове блукання S(n) (n = 1, 2, . . . ) , що породженеточковим процесом S .В бiльшостi робiт, пов’язаних з гiллястими випадковими блуканнями (дляогляду див. [2]), накладається умова M := S(R) < ∞ м.н. В данiй замiтцiми не накладаємо обмежень на величину M , крiм тих, що гарантуютьнадкритичнiсть гiллястого випадкового блукання, тобто те, що популяцiявиживає з додатною ймовiрнiстю. Величина M може бути детермiнованоюабо випадковою, скiнченною або нескiнченою з додатною ймовiрнiстю. Щобзабезпечити надкритичнiсть, ми припускаємо виконаною нерiвнiсть EM > 1у випадку PM <∞ = 1 .Нехай iснує γ > 0 таке, що m(γ) := E
P|u|=1 e
γAu < ∞ . Тут i надалiзапис |u| = n означає, що сумма береться по iндивiдуумам n -го поколiння.Позначимо
Wn :=1
mn(γ)
X|u|=n
eγAu , n = 1, 2, . . .
Послiдовнiсть (Wn,Fn), n = 1, 2, . . . є (невiд’ємним) мартингалом, деFn = σ(S(1), . . . , S(n)) . Покладемо Yi := eγAi
m(γ). Нехай Z -випадкова величина,
розподiл якої задається рiвнiстю
Ef(Z) = EMXi=1
Yif(Yi),
що виконується для довiльної обмеженої борелiвської функцiї f .В роботi [1] була встановлена поведiнка хвоста розподiлу supnWn , в данiйзамiтцi ми узагальнемо цей результат на бiльш широкий клас операторiв.Розглянемо оператор T (див. наприклад [2]), що визначений на множинiмартингалiв (Wn,Fn), n = 1, 2, . . . . Позначимо
T ∗W = supn≥1
TnW,
T ∗∗W = T ∗W ∨ TW,де пiд записом TnW розумiємо, що оператор береться по мартингалам,зупиненим в час n . Тут i надалi вважаємо f∗ := supn≥1 fn для довiльноїпослiдовностi функцiй fn, n = 1, 2, . . . . Будемо казати, що оператор Tзадовольняє умову Aα з параметром α ≥ 1 , якщо виконуються нерiвностi
34
1. E(supn T (|Wn −Wn−1|))α ≤ cE(W ∗)α.
2. E(TW )α ≤ cE(P∞k=1 |Wn −Wn−1|)α.
3. E(supn(|Wn −Wn−1|))α ≤ cE(T ∗W )α.
4. TW ≤ cP∞k=1 |Wn −Wn−1|,
де c - додатнi, але(можливо) рiзнi сталi. Оператор T задовольняє умову B ,якщо iснує 0 < p0 <∞ таке, що для всiх λ > 0 виконуються нерiвностi
1. λp0PTW > λ ≤ c‖W ∗‖p0p0 .
2. λp0PW ∗ > λ ≤ c‖T ∗W‖p0p0 .
Теорема. Нехай T є локальним, квазiлiнiйним, симетричним та вимiрнимоператором, що задовольняє умову B .Нехай iснує b > 1 таке, що E
PMi=1 Y
bi = 1 , E
PMi=1 Y
bi ln+ Yi < ∞ ,
E(PMi=1 Yi)
b < ∞ i оператор T задовольняє умову Aa для довiльного1 ≤ a ≤ b .Якщо lnZ має неарифметичний розподiл, то iснує константа C ∈ (0,∞) така,що
limx→∞
xbPT ∗W > x = C.
Якщо lnZ має арифметичний розподiл з кроком δ , то знайдеться додатнаперiодична з перiодом δ функцiя C(x) така, що для довiльного x ∈ R
limk→∞
e(γk+x)bPT ∗W > eγk+x = C(x).
Зауваження. Наведенi нижче оператори задовольняють умовам теореми
W ∗ = sup1≤n≤∞
Wn,
S(W ) = [
∞Xk=1
(Wn −Wn−1)2]1/2,
s(W ) = [
∞Xk=1
E((Wn −Wn−1)2|Fn−1)]1/2.
Лiтература
[1] A. M. Iксанов, П. А. Негадайлов, Про супремум мартингала, пов’язаногоз гiллястим випадковим блуканням, ТIтаМС, 2006.
[2] D. L. Burkholder, B. J. Davis, R. F. Gundy, Integral inequalities for convexfunctions of operators on martingales, Sixth Berkley Symposium, p. 223-240.
[3] A. M. Iksanov, Elementary fixed points on the BRW smoothing transformswith infinite number of summands, Stoch. Proc. Appl., 114, 27-50, 2004.
КНУ iм. Т. Шевченкаe-mail: [email protected], [email protected]
35
Метод iнтегральних показникiв в оцiнюваннiподаткових ризикiв
Iллiчева Людмила Максимiвна, Україна
Для порiвняльного оцiнювання фiнансового стану пiдприємств,їх надiйностi як клiєнтiв банкiв, можливостi надання кредитiввикористовуються iнтегральнi (комплекснi) показники. Iснує аддитивнаi мультиплiкативна форми iнтегрального показника. Якщо показникимають рiзнi одиницi i непорiвняльнi дiапазони вимiрювань, показникинормують, як правило, переведенням їх у дiапазон вiд 0 до 1. Такожнормують ваговi коефiцiєнти, що отримуються на основi експертнихоцiнок. Застосування такої методологiї для класифiкацiї платникiв накатегорiї уваги можна здiйснювати, розраховуючи iнтегральний показникна основi даних декларацiй. Для порiвняння пiдприємств мiж собою данiдекларацiй представляються вiдносними коефiцiєнтами (факторами) iповиннi мати економiчний змiст. Такий пiдхiд дозволяє аналiзувати тапорiвнювати вплив кожного фактора на значення iнтегрованого показникаi, вiдповiдно, оцiнювати загальний стан платника та його ризиковiсть.Iнтегральний показник ризику декларацiї платника розраховується на основiвiдiбраних та оцiнених експертами вiдносних показникiв (факторiв ризику),їх вагових коефiцiєнтiв i дає можливiсть здiйснити порiвняльний аналiз таранжування платникiв за рiвнем податкового ризику у межах категорiї,до якої вiн вiднесений на основi аналiзу факторiв ризику господарювання(ризики господарювання визначаються на основi iнформацiї, що внесенадо автоматизованої бази даних). Комплексний показник ризику податковоїзвiтностi розраховується на основi значень iнтегральних показникiв ризикузвiтних документiв платника (декларацiй) i дає можливiсть здiйснити бiльшточний порiвняльний аналiз та ранжування платникiв за рiвнем податковогоризику у межах категорiї, до якої вiн вiднесений на основi аналiзуфакторiв ризику госопдарювання. Розрахованi значеня iнтегрованого абокомплексного показника та показник фiскально-фiнансової важливостi даютьзмогу позицiонувати регiони, мiсцевi iнспекцiї та конкретних платникiв нашкалi ризикiв та на часовiй шкалi за наявностi динамiчних рядiв значень.
Кафедра аудиту та економiчного аналiзу,облiково-економiчний факультет,Нацiональна академiя державної податкової служби України,вул. Карла Маркса, 31, Iрпiнь, Київська область, 08200, Українаe-mail: [email protected]
36
Про стiйкiсть випадкових ламаних
Єршов А. В., Україна
Нехай задано послiдовнiсть випадкових векторiв ξn0, . . . , ξnmn ,n = 0, 1, 2 . . . , зв’язаних у кожнiй серiї в ланцюг Маркова;0 = tn0 ≤ tn1 ≤ . . . ≤ tnmn = 1 − деяка послiдовнiсть розбиттiввiдрiзка [0, 1] та λn = max
n[tnk+1 − tnk] → 0 при n→∞ .
Позначимо: ∆tnk = tnk+1 − tnk , ∆ξnk = ξnk+1 − ξnk , Fnk − мiнiмальнаσ -алгебра, вiдносно якої вимiрними є величиниξnj , j ≤ k ; E∆ξnk|Fnk = an(ξnk) ,E(∆ξnk − an(ξnk)∆tnk)(∆ξnk − an(ξnk)∆tnk)T |Fnk = σ2
n(ξnk)∆tnk.Нехай ξn(t) - випадкова ламана з вершинами у точках (tnk, ξnk) :
ξn(t) = ξnk +t− tnk∆tnk
∆ξnk, tnk ≤ t ≤ tnk+1 .
В подальшому будемо використовувати наступнi умови:(H1): σn(x) − невиродженi;(H2): an(x), σn(x) − локально обмеженi ∀n ≥ 1;
(H3): G(x) ∈ C(2)(Rm);
(H4): limN→∞
supn
Psupk|ξnk| > N = 0;
(H5): ∀N > 0 limn→∞
λn sup|x|≤N
˛[σn(x)]−1an(x)
˛2= 0;
(H6): limn→∞
mnPk=0
E˛[σn(ξnk)]−1∆ξnk
˛4χ[−N,N ](ξnk) = 0 ∀N > 0 ,
де χ[−N,N ](ξnk) = 1, якщо ξnk ∈ [−N,N ] i χ[−N,N ](ξnk) = 0 iнакше;
(H7): ∀N > 0 limn→∞
sup|x|≤N
˛mPi=1
G′xi(x)σikn (x)
˛= 0, ∀k = 1, . . . ,m;
(H8): ∀N > 0 limn→∞
sup|x|≤N
˛˛ mPi=1
G′xi(x)ain(x) + 1
2
mPk,i,j=1
G′′xixj(x)σikn (x)σjkn (x)
˛˛ = 0;
(H9): limn→∞
ξn0 = x0 за ймовiрнiстю.
Теорема 1 Нехай виконуються умови (H1)-(H9), тодi ∀ε > 0 ∀δ > 0 iснує
номер n0 такий, що P
(supt∈[0,1]
|G(ξn(t))−G(ξ0)| < ε
)> 1− δ при n > n0.
[1] Кулинич Г.Л. Некоторые предельные теоремы для последовательностицепей Маркова//Теория вероятностей и мат. статистика. − 1975. −Вып.12. − с.77-89.
Київський нацiональний унiверситет iменi Тараса Шевченкаe-mail: [email protected]
37
Использование стандартизованных рисковсмерти при базовой оценке состояния
здоровьяC.С. Карташова (Украина)
Для определения удельного веса влияния антропогенных факторовсреды и наследственной предрасположенности к возникновениюзаболеваний необходима базовая оценка состояния общественногоздоровья. Преобладающее большинство работ, посвященных исследованиюобщественного здоровья, основаны на изучении показателей заболеваемости ираспространенности болезней, инвалидности, донозологической диагностикеболезней в связи с действием факторов окружающей среды. При этом изза различий в доступности медицинской помощи, качестве диагностики,точности регистрации патологических состояний сопоставления могут бытьне вполне корректными. В то же время показатели, основанные на данныхо смерти, отображают реалии, сопоставимы как внутригосударственном,так и на международном уровнях, сведены в соответствующие формыгосударственной статистической отчетности и доступны исследователям.Представляется, что в настоящее время показатели используютсянедостаточно, к тому же не всегда корректно: приводятся только точечныезначения, в то время как методы доказательной медицины требуютинтервальных оценок с указанием соответствующего уровня значимости.
Цель настоящей работы состоит в совершенствовании методов оценкиобщественного здоровья по данным о смерти, распределенным по возрасту,полу и причинам смерти при заданной поло-возрастной структуре cучетом влияния неуправляемых факторов. Изучалась смертность отзлокачественных новообразований в Киевской области. В качестве первичныхматериалов использовались данные о смерти и поло-возрастной структуренаселения Госкомстата Украины. Для усиления статистической мощностивыводов данные по смертности от злокачественных новообразований былиагрегированы по шестилетним периодам.
Учесть влияние факторов неуправляемой природы таких как возраст ипол можно стандартизацией. Выбор метода стандартизации определяется науровне постановки задачи: изучение динамики, статистические сравнения,оценка дополнительного риска, построение прогноза. Стандартизированныйпрямым методом показатель смертности можно интерпретировать какпоказатель смертности стандарта, который вымирает как исследуемоенаселение.
Стандартизованные прямым методом показатели возможно сравниватьчерез статистику отношений SRM = SPM1
SPM2доверительные границы которой
суть:
SRML = SRM1−Zα/2
X ; SRMU = SRM1+Zα/2
X ,
X = SPM1−SPM2√s.e.(SPM1)2−s.e.(SPM2)2
, Zα/2 – квантиль N(0, 1) .
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Применение косвенного метода дает возможность интерпретироватьпоказатель смертности как относительный риск, стандартизированный повозрасту. Анализ риска смерти от злокачественных новообразований поотдельным нозологическим формам среди населения Киевской областипроводился относительно органов дыхания и грудной полости, молочнойжелезы и предстательной железы. Изменения удельного веса отдельнойнозологической формы среди всех случаев исследуемой патологии лучшевсего определять косвенным методом.
При построении прогноза смертности использован кумулятивный метод.Кумулятивный риск – вероятность смерти от исследуемойпатологии за определенный период времени, при условии,что смертность от конкурирующих причин исключена, - сутьKRS = KRS(T ) = 1 − exp(−KIP (T )), где KIP (T ) – накопленнаяинтенсивность смерти за выбранный период [0, ] . Показано, что накопленнаяинтенсивность смерти от онкопатологии за продолжительность жизни(кумулятивный показатель) статистически различна по периодамнаблюдения: если в 1980 - 1985 гг. она составляла 12,29 при 95% ДИ(12,47; 12,10) то в 1986-1991 гг. и 1992-1997 гг. наблюдался рост до14,25 (14,45; 14,05) и 15,38 (15,58; 15,18) соответственно; в 1998 - 2003 гг.произошло значимое снижение до 14,87 при 95% ДИ (15,07; 14,67). В качествеконкурирующих причин к онкопатологии рассматривались все причинысмерти за исключением злокачественных новообразований. Наибольшиетемпы роста пришлись на период 1986-1991 гг., при этом вероятность смертиза ожидаемую продолжительность жизни возросла в 1,23, а накопленнаяинтенсивность смерти – в 1,33 раза. Уменьшение темпов роста в период1992-1997 гг. обусловило более умеренный рост вероятности смерти от этихпричин в 1998-2003 гг. (на 3%), а интенсивности кумулятивной смертностина 4% по отношению к 1986-1991 гг. Выявлено, что снижение риска смертиот злокачественных новообразований за ожидаемую продолжительностьжизни не изменит в течение ближайших 6 лет динамику общей смертности,которая имеет тенденции к росту, но при более низких темпах, чем ранее.
Проведенный анализ свидетельствует о том, что корректноеиспользование показателей причинно-специфической и общей смертности,стратифицированных по полу, стандартизованных по возрасту снадлежащими характеристиками дает возможность базовой характеристикиобщественного здоровья. Можно статистически корректно ранжироватьтерритории по уровню общей и причинно-специфической смертности. Всевышеизложенное является основанием для более широкого применениястандартизованных рисков смерти и показателей смертности в оценкеобщественного здоровья.
Кафедра статистики и эконометрии,учетно-финансовый факультет, КНТЭУ,02156 Киев-156, ул.Киото, 19e-mail: [email protected]
39
Про множину розв’язкiв рiвняння νi(x) = x
Котова О.В., Україна
Будь-яке число x ∈ [0; 1] можна подати у виглядi
x =
∞Xk=1
s−kαk, (1)
де s ∈ N , s ≥ 2 , αi ∈ 0, 1, ..., s− 1 .Вираз (1) називається s -адичним розкладом (представленням) числа x ,
αk = αk(x) — k -тою цифрою s -адичного представлення числа x .Позначимо через Ni(x, k) кiлькiсть цифр ” i ” в s -адичному розкладi числа
x до k -го мiсця включно. Якщо границя limk→∞
Ni(x,k)k
iснує, то її значення
νi(x) називається частотою цифри ” i” в s -адичному представленнi числаx .
Добре вiдома теорема Лебега стверджує, що майже всi x ∈ [0; 1]мають частоти всiх цифр рiвнi 1
s, а множина чисел, якi не мають частоти
жодної цифри, є суперфрактальною (тобто має мiру Лебега 0 i розмiрнiстьХаусдорфа-Безиковича рiвну 1) [1]. Зрозумiло, що функцiя частоти νi(x)цифри x є всюди розривною.
В доповiдi представляються результати дослiджень iснування тапотужностi множини розв’язкiв частотного рiвняння νi(x) = x в трiйковiйсистемi числення.
Теорема 1 Якщо εn — довiльна нескiнченна послiдовнiсть нулiв таодиниць, причому
sn = (n+ 1)! + 1, en = sn+1 − sn − 1 = (n+ 2)!− (n+ 1)!− 1,
то число
x =
∞Xi=1
εi3si
+
∞Xj=1
ejXi=1
βij3sj+i
,
де
βin = [(sn + i)xn]− [(sn + i− 1)xn], xn =
nXi=1
εi3si
+
n−1Xj=1
ejXi=1
βij3sj + 1
,
є розв’язком частотного рiвняння
ν(3)1 (x) = x. (2)
Наслiдок 3 Рiвняння (2) має континуальну множину розв’язкiв.
Лiтература
[1] Працьовитий М.В. Фрактальний пiдхiд у дослiдженнях сингулярнихрозподiлiв. — Київ: Вид-во НПУ iменi М.П.Драгоманова, 1998. — 296 с.
НПУ iменi М.П.Драгоманова, Київ, Україна
40
Тополого-метричнi властивостi однiєї сiм’їкомплекснозначних випадкових величин
Коваленко В.М., Україна
В роботi [1] М.В. Працьовитий ввiв класифiкацiю сингулярних розподiлiвдiйснозначних випадкових величин в залежностi вiд тополого-метричнихвластивостей їх спектрiв. Такий самий пiдхiд був ним запропонований встаттях [4], [2] i для комплекснозначних випадкових величин. Дослiдженнятополого-метричних властивостей спектрiв конкретних випадкових величинє самостiйною i часто складною задачею (див. наприклад [1], [3]).
Нами розглядається будова та метричнi властивостi спектра наступноїсiм’ї комплекснозначних випадкових величин
ξ =
∞Xm=1
rmεηm , (1)
де∞Pm=1
rm – збiжний знакододатний ряд, в якому rm > rm+1,∀m ∈ N ;
εk = ei2πk
n – корiнь n -го степеня з одиницi (k = 0, n− 1, n ≥ 3, n ∈ N) ;ηm –дискретнi випадковi величини, що набувають значень 0, 1, ..., n − 1 зймовiрностями p0m, p1m, ...,
p(n−1)m вiдповiдно,n−1Pl=o
plm = 1 ∀m ∈ N.
Розподiл випадкової величини ξ визначається трьома параметрами:показником степеня n , послiдовнiстю rm , та матрицею ймовiрностейP = ||plm|| .
Iз результатiв статтi [4] випливає, що розподiл ξ є: а) дискретним, тодi i
тiльки тодi, коли∞Qm=1
maxlplm > 0; б) неперервним, тодi i тiльки тодi, коли
∞Qm=1
maxlplm = 0.
Набагато складнiшою i повнiстю ще не розв’язаною є задача знаходженнянеобхiдних i достатнiх умов сингулярностi та абсолютної неперервностiрозподiлу. Однiєю з пiдзадач є дослiдження топологiчних та метричнихвластивостей спектра випадкової величини.
На тополого-метричнi властивостi Sξ при фiксованих n та rmвпливає розмiщення нулiв в матрицi ||plm|| . Для дослiдження спектравипадкової величини, яка має нулi в матрицi ймовiрностей, треба знати йоговластивостi для випадку, коли всi ймовiрностi додатнi. Саме цей випадок миi розглядаємо.
Доповiдачем були доведенi наступнi твердження.
41
Теорема 1. Якщо для всiх натуральних k виконується нерiвнiсть
rk ≤sin(2k0 + 1)π
n
sin πn
∞Xm=k+1
rm, (2)
то спектр випадкової величини ξ є зв’язною множиною.
Теорема 2. Якщо rm = qm , де q ≤ sin πn
sin πn
+sin(2k0+1)π
n
, k0 –цiле число з
промiжка [n4− 1; n
4) , то спектр випадкової величини ξ є самоподiбною
множиною з розмiрнiстю Хаусдорфа-Безиковича рiвною − lnnln q
.
Теорема 3. Якщо для всiх натуральних k виконується нерiвнiсть
rk >sin(2k0 + 1)π
n
sin πn
∞Xm=k+1
rm, (3)
де k0 –цiле число з промiжка [n4− 1; n
4) , то двовимiрна мiра Лебега спектра
випадкової величини ξ дорiвнює наступнiй границi
λ(Sξ) = limk→∞
1
2sin
2π
nnk+1(
∞Xm=k+1
rk)2.
Лiтература
[1] Працьовитий М.В. Фрактальний пiдхiд у дослiдженнях сингулярнихрозподiлiв /Вид-во НПУ iменi М. П. Драгоманова, Київ–1998.–296 с.
[2] Працьовитий М.В., Чумак М.Є. Фрактальнiсть i канторовiсть роз-подiлу однiєї комплекснозначної випадкової величини типу Джессена-Вiнтнера//Науковi записки НПУ iменi М.П.Драгоманова.- Київ: НПУiменi М.П.Драгоманова, 1999.- с.244-249.
[3] Турбин А.Ф., Працевитый Н.В. Фрактальные множества, функции,распределения.-К.: Наук. думка, 1992.-208с.
[4] Школьний О.В., Працьовитий М.В. Один клас сингулярних комплекс-нозначних випадкових величин типу Джессена-Вiнтнера//Укр. мат.журн.–1997.–т.49, 12.–с.1653-1660.
[5] Hutchinson J.E.(1981) Fractals and self-similarity, Indiana Univ. Math. J.,30, 713-747.
НПУ iменi М.П.Драгоманова, Київ, Україна
42
Iнтегральнi многовиди стохастичнихдиференцiальних систем Iто
Андрiй Креневич, Україна
На ймовiрнiсному просторi (Ω,F, P ) з фiльтрацiєю Ft, t ∈ R ⊂ Fрозглядається система стохастичних диференцiальних рiвнянь Iто
dx = f(t, x)dt, (1)
dy = (B(t)y + g(t, x, y))dt+ σ(t, x, y)dWt, (2)де f(t, x) ∈ C(R×Rn) – n -вимiрна функцiя, g(t, x, y), σ(t, x, y)∈ C(R × Rn × Rm) – m -вимiрнi функцiї. Wt – стандартний скалярнийвiнерiвський процес, визначений для t ∈ R на ймовiрнiсному просторi(Ω,F, P ) i узгоджений з фiльтрацiєю Ft, t ∈ R .
Означення 1 Випадковий процес ϕ(t, x) будемо називати iнтегральниммноговидом системи (1), (2), якщо для довiльного розв’язку x(t) системи(1) процес y(t) = ϕ(t, x(t)) є розв’язком системи (2).
Означення 2 Випадковий процес ϕ(t, x) будемо називати обмеженим всередньому квадратичному iнтегральним многовидом системи (1), (2),якщо ϕ(t, x) є iнтегральним многовидом системи (1),(2), причому iснуєρ > 0 , таке, що для довiльних t ∈ R, x ∈ Rn виконується нерiвнiстьE|ϕ(t, x)|2 ≤ ρ2 .
Теорема 1 1)Нехай для системи (1) виконуються умови єдиностiрозв’язку. 2)Нехай Y (t, τ) – матрицант системи
dy/dt = B(t)y
для довiльних t, τ задовольняє умову
‖Y (t, τ)‖ ≤ Ke−γ(t−τ),
де K, γ – деякi додатнi сталi незалежнi вiд t, τ .3)Iснують додатнi сталi L,M такi, що для довiльних t, x1, x2, y1, y2 функцiїg, σ – задовольняють умовам
|g(t, x1, y1)− g(t, x2, y2)|+ |σ(t, x1, y1)− σ(t, x2, y2)| ≤ L(|x1 − x2|+ |y1 − y2|),
|g(t, x1, 0)|+ |σ(t, x1, 0)| ≤M.
4)Сталi K, γ, ρ, L,M пов’язанi спiввiдношеннями
K
γ(Lρ+M) +K
r1
γ(L2ρ2 +M2) ≤ ρ,KL
„1
γ+
1√2γ
«< 1.
Тодi система (1), (2) має обмежений в середньому квадратичномуiнтегральний многовид, який можна знайти використовуюючи методпослiдовних наближень.
Київський нацiональний унiверситет iм.Тараса Шевченкаe-mail: [email protected]
43
Про компактнiсть розв’язкiв стохастичногодиференцiального рiвняння при
нерегулярнiй залежностi вiд параметруКулiнiч Г.Л., Очкур О.В., Україна
Нехай при кожному значеннi параметра α > 0 iснує сильний розв’язокξα(t) стохастичного диференцiального рiвняння
dξα(t) = aα(t, ξα(t))dt+ σα(t, ξα(t))dηα(t), (1)
де aα(t, x), σα(t, x), t ∈ [0, 1], x ∈ (−∞,∞) - неперервнi дiйснi функцiї;ηα(t) - при кожному α неперервний з ймовiрнiстю 1 iнтегрований з квадратоммартингал з характеристикою 〈ηα(t)〉 = [1 + ψ(α)]t + λα(t) , де ψ(α) –
невипадкова величина i ψ(α) → 0, E1
V0λα(t) → 0 при α → 0,
1
V0λα(t) -
варiацiя процесу λα(t) .Ранiше питання про граничну поведiнку розв’язку (1), в якому
коефiцiєнти явно не залежать вiд t дослiджували в [1], а в роботi [2] − колиηα(t) - сiм’я вiнерiвських процесiв.
Теорема 1 Нехай iснують функцiї aα(x), σα(x) , такi що:1. 0 < [f ′α(x)σα(x)]
2 ≤ Lˆ1 + |fα(x)|2
˜,
де fα(x) = c(1)α
xR0
exp
−2
uR0
aα(v)σα(v)
dv
ffdu+ c
(2)α , c
(1)α , c
(2)α - сiм’я певних сталих;
2.
"1 + sup
|fα(x)|≤N
˛aα(x)σα(x)
˛#×»
1R0
qkNα(t)dt+ sup
0≤t≤1q2Nα
(t)E1
V0λα(t) + ψ(α) + E
1
V0λα(t)
–→ 0
при α → 0 для довiльного N > 0, k = 1, 2, деqNα(t) = sup
|fα(x)|≤N
˛σα(t,x)−σα(x)
σα(x)
˛;
3.1R0
sup|fα(x)|≤N
˛aα(t,x)−aα(x)
σα(x)
˛dt→ 0 при α→ 0 для довiльних N > 0;
4. E [fα(ξα(0))]2 ≤ L.Тодi сiм’я процесiв fα(ξα(t)) слабко компактна.
[1] Кулинич Г.Л. Предельная теорема для одномерных стохастическихдифференциальных уравнений при нерегулярной зависимостикоэфициентов от параметра//Теория вероятностей и математическаястатистика. – 1976. – в. 15. – с. 99 - 113.
[2] Алмазов М., Кулинич Г.Л. Предельные теоремы для одномерныхнеоднородных стохастических диффузионных уравнений принерегулярной зависимости коефициентов от параметра// Укр.мат. журнал. – 1990. – т. 42, 1. – с. 435 - 443.
Київський нацiональний унiверситет iменi Тараса Шевченка
44
Функцiональна центральна граничнатеорема для бакстерiвських сум гауссового
випадкового поля з однорiдними приростами
Олександр Курченко, Україна
У статтi [1] доведена функцiональна центральна гранична теорема дляпослiдовностi випадкових полiв, побудованої за допомогою нормованих сумнелiнiйних функцiй для мультиiндексної послiдовностi серiй випадковихвеличин, якi у кожнiй серiї мають сумiсний гауссовий розподiл з нульовимматематичним сподiванням. Ця послiдовнiсть випадкових полiв слабкозбiгається у просторi Скорохода D([0, 1]d) до випадкового поля Ченцова.Застосування цих результатiв для бакстерiвських сум гауссового випадковогополя з однорiдними приростами дозволяє отримати наведену нижче теорему.
Нехай X(t), t ∈ Rd, d ≥ 2 — гауссове випадкове поле з нульовимматематичним сподiванням та кореляцiйною функцiєю r(t, s), t, s ∈ Rd . Дляt = (t1, . . . , td), h = (h1, . . . , hd), hi > 0, 1 ≤ i ≤ d,
X([t, t + h]) =nP
i1=0
· · ·nP
id=0
(−1)i1+...+id · X(t1 + i1h1, . . . , td + idhd) — прирiст
випадкового поля X на d -вимiрному паралелепiпедi [t, t+h] . Через λn = qпозначимо рiвномiрне розбиття [0, 1]d на nd конгруентних d -вимiрнихпаралелепiпедiв.Покладемо
Sn =Pq∈λn
G
„X(q)√VarX(q)
«, d2n = VarSn, Vn(t) = 1
dn
Pq∈λnq⊂[0,d]
G
„X(q)√VarX(q)
«,
t ∈ [0, 1]d , де G — многочлен Чебишова-Ермiта з парним iндексом.
Теорема 1 Нехай X(t), t ∈ Rd — гауссове випадкове поле знульовим математичним сподiванням, однорiдними приростами,кореляцiйною функцiєю r(s, t), s, t ∈ Rd , G — многочленЕрмiта степеня 2m i виконанi наступнi умови 1) кореляцiйнафункцiя r(s, t) 2d раз неперервно диференцiйована при t 6= s i∃L, θ ≥ 0 :
˛∂r(s,t)
∂t1...∂td∂s1...∂sd
˛≤ L
‖s−t‖θ , де ‖ · ‖ — евклiдова норма в
Rd ; 2) ∃C,α > 0∀q = [(0, . . . , 0), (h, . . . , h)] ⊂ [0, 1]d : EX2(q) ≥ Chα ;3) θ > d
2mi α ≥ 2d − θ ; 4) ∀s, t ∈ [0,+∞)d iснує скiнченна границя
limh→0+
E(X([ht,ht+y])X([hs,hs+y]))
EX2([0,y])=: g(t− s) , де y = (h, . . . , h) ∈ Rd ;
Тодi послiдовнiсть розподiлiв випадкових полiв Vn(t), t ∈ [0, 1]d : n ≥ 1у просторi Скорохода D([0, 1]d) слабко збiгається до розподiлу випадковогополя Ченцова W (t), t ∈ [0, 1]d .
1. Курченко О. О. Збiжнiсть у просторi D([0, 1]d) однiєї послiдовностiвипадкових полiв // Теорiя ймовiрностей та математична статистика. — 2001.— Вип. 64. — С. 82–91.
45
Випадковi ланцюговi дроби та їх фрактальнiвластивостi
Д.В. Кюрчев, Україна
Нехай ξ - випадкова величина, представлена елементарним ланцюговимдробом
ξ =1
η1 +1
η2 +1
η3 + . . .
= [η1, η2, η3, . . .], (1)
де ηk - незалежнi однаково розподiленi випадковi величини, що приймаютьнатуральнi значення 1, 2, . . . , k, . . . вiдповiдно з iмовiрностями p1 ,p2 ,. . . ,pk, . . . , де pk ≥ 0 для всiх k ∈ N ,
∞Xk=1
pk = 1.
Пiд фрактальними властивостями [2] розподiлу випадкової величинирозумiтимемо фрактальнi властивостi його спектра та носiя. Звичайно,фрактальними (у вузькому смислi) властивостями спектр володiє лише увипадку канторовостi.
Позначимо E = E(A) - множина дiйсних чисел x ∈ (0; 1] , розклад яких велементарний ланцюговий дрiб x = [a1, a2, a3, . . .] мiстить лише елементи aiз даної множини A ⊂ N .
Тодi вiдомо [2], що спектр розподiлу випадкового ланцюгового дробу (1) єзамиканням множини типу E = E(A) i буде множиною канторiвського типуза умови A 6= N .
Нехай α ≥ 0 - дiйсне число. Тодi α -мiрна мiра Хаусдорфа обмеженоїмножини E метричного простору (M,ρ) визначається як
Hα(E) = limε→0
inf
d(Ej)≤ε
(Xj
dα(Ej)
)!,
де iнфiмум береться по всiх не бiльш нiж зчисленних ε -покриттях Ejмножини E , d(Ej) = supρ(x, y) : x ∈ Ej , y ∈ Ej .
Означення 1 Дiйсне число
α0(E) = sup α : Hα(E) = ∞ = inf α : Hα(E) = 0
називається розмiрнiстю Хаусдорфа-Безиковича множини E [2].
Але при обчисленнi розмiрностi Хаусдорфа-Безиковича iнодi зручновикористовувати бiльш вузькi класи покриттiв за умови, що вони даютьоднакове значення розмiрностi.
46
У доповiдi пропонується доведення еквiвалентностi використанняцилiндричних вiдрiзкiв ланцюгового представлення - замкнених iнтервалiв зкiнцями [a1, a2, . . . , an] i [a1, a2, . . . , an+1] в якостi класу покриттiв множиниE(A) при знаходженнi її розмiрностi Хаусдорфа-Безиковича.
Це дало можливiсть отримати оцiнки розмiрностi Хаусдорфа-Безиковичамножини E(A) = Em1,mk , де A = m1,m2, . . . ,mk , mi ∈ N , i = 1, k .
Теорема 1 Розмiрнiсть Хаусдорфа-Безиковича множини Em1,mk
задовiльняє нерiвностям:
ln k
2 lnmk +
pm2k + 4
2
≤ α0(Em1,mk ) ≤ln k
2 lnm1 +
pm2
1 + 4
2
. (2)
Наслiдок 4 Якщо A = m,m+1, . . . , cm , де c - константа, причому c ≥ 2 ,c ∈ N , то α0(E(A)) → 1
2при m→∞ .
Наслiдок 5 A = m,m + 1, . . . ,m + c , де c - константа, причому c ≥ 2 ,c ∈ N , то α0(E(A)) → 0 при m→∞ .
Зауваження 1 T.W. Cusick [3] i G. Ramharter [5] запропонували формулудля обчислення розмiрностi Хаусдорфа-Безиковича множини E = E(A) :
α0(E(A)) =1
2limN→∞
lnSN (A)
lnN, (3)
де SN (A) — кiлькiсть наборiв a1, a2, . . . , an , ai ∈ A , для яких знаменникипiдхiдних дробiв qn = q(a1, a2, . . . , an) (n = 1, 2, . . .) не перевищують N .
Використовуючи формулу (3), можна отримати оцiнки розмiрностi,аналогiчнi (2).
Лiтература
[1] Працьовитий М. В. Фрактальний пiдхiд у дослiдженнях сингулярнихрозподiлiв. — Київ: Вид-во НПУ iменi М. П. Драгоманова, 1998. — 296 с.
[2] Хинчин А.Я. Цепные дроби. — М.: Наука, 1978. —116c.[3] Cusick T. W. Continuants with bounded digits // Mathematika. — 1977. —
24. — P.166-172.[4] Kinney J.R., Pitcher T.S. The dimension of some sets defined in terms of f -
expansions // Z. Wahrscheinlihkeitstheorie verw. Geb. — 1966. — 4. — P.293-315.
[5] Ramharter G. Some metrical properties of continued fractions // Mathe-matika. — 1983. — 30. — P.117-132.
Нацiональний педагогiчний унiверситет iм. М.П. Драгоманова,кафедра вищої математики,вул. Пирогова, 9, м. Київ 01030e-mail: [email protected]
47
Дискретнi функцiї розподiлу зi складноюлокальною будовою та їх фрактальнi
властивостiД.В. Кюрчев, О.Ф. Георгiєва, Україна
Як вiдомо, кожне рацiональне число x ∈ (0, 1] можна представити увиглядi
x =1
a1 +1
a2 +. . . +
1
ak
= [a1, a2, . . . , ak],
а кожне iррацiональне - у виглядi x = [a1, a2, a3, . . .] , який називаєтьсярозкладом x в елементарний (неперервний) ланцюговий дрiб, де ak = ak(x)(k ∈ N ) - натуральнi числа, якi називають його елементами ("цифрами").
Розглянемо функцiю y = f(x) на вiдрiзку [0; 1] . Її аргумент xпредставимо ланцюговим дробом x = [a1, a2, . . . , ak, . . .] , x ∈ (0; 1] .Значення функцiї y = f(x) також представимо ланцюговим дробомy = [b1, b2, . . . , bk, . . .] , де bk набувають натуральних значень, є функцiямивiд y (а отже, i вiд x) i визначаються рiвнiстю
bk =
kXi=1
ai.
Покладемо y(0) = 0 . Функцiя y = f(x) є зростаючою на вiдрiзку [0; 1] iнеперервною в кожнiй iррацiональнiй точцi вiдрiзку [0; 1] .
Визначимо фунцiю F (x) так само як i f(x) , за винятком рацiональнихточок виду x0 = [a1, a2, . . . , am] , m - парне, в яких покладемо
F (x0) = limx→x0−0
f(x) = [a1, a1 +a2, . . . , a1 +a2 + . . .+am− 1, a1 +a2 + . . .+am].
Теорема 1 Функцiя F (x) є дискретною функцiєю розподiлу без iнтервалiвпостiйностi.
Зазначимо серед диференцiальних властивостей функцiї F (x) такi:1) Похiдна фунцкцiї y = F (x) майже всюди рiвна нулю.2) Носiй NF функцiї розподiлу F (x) має мiру Лебега нуль.3) В точках з обмеженими елементами ланцюгового представлення
похiдна функцiї розподiлу F (x) дорiвнює нулю.
Теорема 2 Для того, щоб iррацiональна точка x0 = [a1, a2, . . . , ak, . . .] нале-жала носiю NF , достатньо щоб ak > (k1+ε − 1)(a1 + . . .+ ak−1) для всiх kпочинаючи з деякого номеру k0 , де ε > 0 .
Наслiдок. Розподiл, що задається функцiєю F (x) , є фрактальним.
48
Множиною значень E(F ) функцiї розподiлу F (x) є континуальнамножина всiх ланцюгових дробiв вiдрiзку [0;1], елементи яких утворюютьзростаючi послiдовностi натуральних чисел, тобто множина
x : ak+1(x) > ak(x) ∀k ∈ N,
доповнена рацiональними точками виду x = [a1, a2, . . . , am−1, am] , деak+1 > ak , k = 1,m , m - непарне.
Множина E(F ) майже замкнена, тобто мiстить всi свої граничнi точкиза винятком рацiональних точок виду x = [a1, a2, . . . , am−1, am] , ak+1 > ak ,k = 1,m (m - парне), обмежена, без iзольованих точок. Легко бачити, щоE(F ) є нiде не щiльною множиною мiри Лебега нуль.
Позначимо CN - замикання множини значень E(F ) функцiї F .Розглянемо фрактальнi властивостi цiєї множини та деяких її пiдмножин.
Теорема 3 Позначимо множиниCM = x : 0 < ak(x)− ak−1(x) ≤M ∀k = 2, 3, . . . ,C∞ = x : ak(x)− ak−1(x) →∞ (k →∞) ,E∞ = x : ak(x) →∞ (k →∞) .Тодi:1) Множина CM є аномально фрактальною.2) Розмiрнiсть Хаусдорфа-Безиковича множини C∞ дорiвнює 1
2.
3) Розмiрнiсть Хаусдорфа-Безиковича множини CN дорiвнює 12.
Наслiдок. Функцiя F (x) зберiгає розмiрнiсть Хаусдорфа-Безиковича довi-льної непорожньої пiдмножини множини E∞ .
Лiтература
[1] Лисовик Л.П. Применение конечных преобразователей для заданияфрактальных кривых // Кибернетика и системный анализ. — 1994. — 3. — С.11-22.
[2] Працьовитий М. В. Фрактальний пiдхiд у дослiдженнях сингулярнихрозподiлiв. — Київ: Вид-во НПУ iменi М. П. Драгоманова, 1998. — 296 с.
[3] Хинчин А.Я. Цепные дроби. — М.: Наука, 1978. —116c.[4] Albeverio S., Pratsiovytyi M., Torbin G. Fractal probability distributions and
transformations preserving the Hausdorff-Besicovitch dimension // Ergod. Th.and Dynam.Sys. — 2004. — 24. — P.1-16.
Нацiональний педагогiчний унiверситет iм. М.П. Драгоманова,вул. Пирогова, 9, м. Київ 01030e-mail: [email protected]
Промислово-економiчний коледж,Нацiонального Авiацiйного унiверситету,вул. Метробудiвська, 5а, м. Київ 03065e-mail: [email protected]
49
Робастне оцiнювання по спостереженнях здомiшкою
Анастасiя Лодатко, Україна
Розглядається вибiрка (ξi:N , i = 1...N) з двокомпонентної сумiшi зiзмiнними концентрацiями. ξi:N - незалежнi випадковi величини з функцiєюрозподiлу
Pξi:N < x = wi:NH1(x, θ) + (1− wi:N )H2(x),
де H1−розподiл основної компоненти, який вiдомий з точнiстю до параметруθ ∈ Θ ⊆ R , H2−розподiл домiшки (повнiстю невiдомий), x ∈ R , wi:N -концентрацiя i−тої компоненти у сумiшi. Концентрацiї змiнюються з кожнимновим спостереженням i вважаються вiдомими.
Щоб оцiнити θ , розглянемо теоретичну медiану:
M(θ) = medH1(x, θ) = supt : H1(t, θ) ≤ 1
2.
Хорошею оцiнкою H1 є зважена емпiрична функцiя розподiлу
H1(x) =1
N
NXi=1
a1i:N1ξi ≤ x,
де aki:N = NNP
i=1(wi:N−w)2
2Pm=1
(−1)m+kγkm:Nwmi:N - мiнiмакснi ваговi коефiцiєнти,
γkm:N - km-мiнор матрицi
ΓN =1
NWTW, w =
1
N
NXi=1
wi:N .
На роль оцiнки для медiани H1 використаємо медiану H1(x) :
yN = med H1(x) = supt : H1(t) ≤ 1
2.
Прирiвнюючи теоретичну та емпiричну медiани, отримуємо медiаннуоцiнку параметру:
θ = M−1(yN ).
Доведена консистентнiсть та асимптотична нормальнiсть побудованоїоцiнки.
Кафедра теорiї ймовiрностей та математичної статистики,механiко-математичний факультет,Київський нацiональнiй унiверситет iменi Тараса Шевченка,вул. Володимирська, 64, 01033 Київ, Українаe-mail: [email protected]
50
Про випадковi булевi матрицiВолодимир Масол, Україна.
У доповiдi пропонується огляд результатiв дослiдження ранга випадковоїбулевої матрицi, а також наступна теорема.
Нехай νn - число нетривiальних розв’язкiв системи лiнiйних однорiднихрiвнянь
A−→X =
−→0 , (1)
де A = ‖atj‖ , t = 1, ..., T , j = 1, ..., n - (T × n) - матриця коефiцiєнтiвсистеми (1) над полем GF (2) ; елементи atj - незалежнi випадковi величини,що приймають значення 0 або 1 (0, 1 ∈ GF (2)) з iмовiрнiстю P atj = 1 = p(t = 1, ..., T , j = 1, ..., n). Позначимо через L(r0, s0) подiю, яка полягає втому, що матриця A має r0 нульових рядкiв та s0 нульових стовпцiв.
Теорема 1 Нехай виконуються умови n− T = m0, де m0 фiксоване число;lnn−ωn
≤ p ≤ 1− lnn−ωn
, де
ω = ρlnn, ρ = const, 0 ≤ ρ <1
2. (2)
Тодi Pνn = 2k − 1/L(0, 0) = 0 для k < µ, µ = max(m0, 0) ,limn→∞
Pνn = 2k − 1/L(0, 0) = π(k;µ) для k ≥ µ ,
де π(k;µ) = 2−(k−µ)k
k−µQt=1
(1− 2−t)
ff−1 ∞Qt=k+1
(1− 2−t) .
Зауваження 1 Доведення теореми може бути побудовано на перевiрцiумов теореми 1 роботи [1]. На перевiрцi умов цiєї ж теореми 1 можнадослiдити також iмовiрнiсть Pνn = 2k − 1/L(r0, s0)(n → ∞) дляслабозаповненої матрицi A , тобто матрицi для якої p = lnn−ω
n, де ω
задовiльняє (2). Тут r0 та s0 - фiксованi цiлi невiд’ємнi числа.
Зауваження 2 В роботi [2] випадкова величина νn (n → ∞) дослiдженапри умовi на рядки та стовпцi, яка включає, зокрема, умову L(0; 0) .
Лiтература
[1] Масол В.И., Расширение области инвариантности для случайныхбулевых матриц. Кибернетика, 1980, 3, 125–128.
[2] Cooper C., On the rank of random matrices. Random Structures and Algo-rithms, 2000, v.16, 2, 209–232.
Кафедра теорiї ймовiрностей та математичної статистики,механiко-математичний факультет,Київський нацiональнiй унiверситет iменi Тараса Шевченка,проспект Академiка Глушкова, 6, Київ, 03127, Українаe-mail: [email protected]
51
Про одне застосування граничногорозподiлу рангу випадкової розрiдженої
булевої матрицi
Володимир Масол, Україна
Свiтлана Поперешняк, Україна
Нехай задана матриця A , A = ‖aij‖ , над полем GF (2) , що складаєтьсяз двох елементiв, з вiдомим розподiлом незалежних випадкових величин aij ,i = 1, T , j = 1, n , T/n/ - число рядкiв /стовпцiв/ матрицi A . Розглянемонеоднорiдну систему лiнiйних рiвнянь
AX = B, (1)
де елементи T × n матрицi A = ‖aij‖ незалежнi випадковi величини i дляi = 1, T , j = 1, n
P aij = 1 = 1− P aij = 0 =lnn+ xij
n, (2)
де
|xij | ≤ c, c = const, i = 1, T , j = 1, n. (3)
Вектор-стовпець B = (b1, . . . , bT ) не залежить вiд A, випадковiвеличини b1, . . . , bT незалежнi та приймають значення 0 i 1 з ймовiрностями
P bi = 0 =1
2(1 + εi(n)) , i = 1 , 2, . . . , T.
X = (x1, x2, . . . , xn) - n- вимiрний вектор-стовпець невiдомих величин.Позначимо Pn,T ймовiрнiсть того, що система (1) є сумiсною системою.Теорема 1. Нехай для усiх n ≥ n0 > 1 0 < T
n≤ 1− log2 lnn
(lnn)q , q = const ,0 < q < 1, i виконуються умови (2) та
ε(n) ≤ β < 1, β = const.
де
ε(n) = max1≤k≤T
|εk(n)| .
Тодi має мiсце
−α2f (n)− ε(n) ≤ Pn,T − exp
„−λ
2
«≤ α1f (n) + ε(n),
52
де f(n) = 4 (1 + δ) eec ln4 nn(ln lnn)2
, α1 = 1 + 11−ε(n)
, α2 = 1 + ∆(n) + 11+ε(n)
,
λ = 1n
TPi=1
exp
(− 1n
nPj=1
xij
), δ > 0, δ = const,∆(n) → 0 при n→∞ .
Далi припустимо, що
P aij = 1 = 1− P aij = 0 =lnT + xij
T, (4)
де xij задовольняє умову (3). Покладемо
P bi = 0 =1
2(1 + εi(T )) , i = 1 , 2, . . . , T. (5)
Нехай також
λ1 =1
T
nXi=1
exp
(− 1
T
TXj=1
xij
).
Теорема 2. Нехай для усiх n ≥ n0 > 1 1 + log2 lnn
(lnn)q ≤ Tn
< ∞ ,q = const, 0 < q < 1, i виконуються умови (3)-(5) та
ε(T ) ≤ γ < 1, γ = const,
де max1≤k≤n
|εk(T )| = ε(T ) .
Тодi має мiсце
−α∗2f (T )− ε(T ) ≤ Pn,T − exp
„−λ1
2
«≤ α∗1f (T ) + ε(T ),
де f(T ) = 4 (1 + δ) eec ln4 TT (ln lnT )2
, α∗1 = 1 + 11−ε(T )
, α∗2 = 1 + ∆(T ) + 11+ε(T )
,
∆(T ) → 0 при T →∞.
[В. I. Масол]Кафедра теорiї ймовiрностей та математичної статистики,механiко-математичний факультет,Київський нацiональнiй унiверситет iменi Тараса Шевченка,проспект Академiка Глушкова, 6, Київ, 03127, Українаe-mail: [email protected]
[C. В. Поперешняк]Кафедра теорiї ймовiрностей та математичної статистики,механiко-математичний факультет,Київський нацiональнiй унiверситет iменi Тараса Шевченка,проспект Академiка Глушкова, 6, Київ, 03127, Українаe-mail: [email protected]
53
Асимтотика математичного сподiваннячисла хибних розв’язкiв системи нелiнiйних
рiвнянь над полем GF(4)
Володимир Масол, УкраїнаЛюдмила Ромашова, Україна
Розглянемо систему рiвнянь
nXk=1
X1≤j1<...<jk≤n
a(i)j1...jk
xj1 ...xjk = bi (i = 1, 2, ..., N) (1)
над скiнченним полем, яке складається з чотирьох елементiв, при умовi (А):коефiцiєнти a
(i)j1...jk
(1 ≤ j1 ≤ ... ≤ jk ≤ n, k = 1, 2, ..., n, i = 1, 2, ..., N) -незалежнi випадковi величини з розподiлом Pa(i)
j1...jk= T = 1 − 3pik ,
T ∈ GF(4) , T = 0 i Pa(i)j1...jk
= T = pik , при T ∈ GF (4) , T 6= 0 ; елементиbi (i = 1, 2, ..., N) - результат пiдстановки в лiву частину (1) фiксованого n-вимiрного вектора x0 , компоненти якого належать полю GF (4) .
Позначимо νn число хибних розв’язкiв системи (1), тобто число розв’язкiвсистеми (1), якi вiдмiннi вiд x0 .
Нехай ρT (n) кiлькiсть компонент вектора x0 , якi дорiвнюють T , деT ∈ GF (4) , T 6= 0 .
Теорема. Нехай виконуються умови: (А);ρT (n) = ρTn , T ∈ GF (4) , T 6= 0 , 0 <
PT 6=0
ρT ≤ 1 ;
14− 1
4δ2 ≤ pi2 ≤ 1
4+ 1
4δ2 , δ2, δ2 - фiксованi числа, δ2 ∈ [0, 1] , δ2 ∈ [0, 1
3] ,
14− 1
4δk ≤ pik ≤ 1
4+ 1
4δk , δk, δk - фiксованi числа, δk ∈ [0, 1) , δk ∈ [0, 1
3] ,
(i = 1, 2, ..., N , k = 1, 3, 4, ..., n) i N = n+m , m - фiксоване число довiльногознаку.
Тодi Eνn → 4−m при n→∞ .Аналогiчнi дослiдження для поля, що складається з двох елементiв,
проведенi в роботi [1].
Лiтература
[1] Masol V. I. Moments of the number of solution of a system of random Booleanequations. - Random Operators and Stochastic Equations, 1993, v. 1, 2, p.171 - 179.
Кафедра теорiї ймовiрностей та математичної статистики,механiко-математичний факультет,Київський нацiональний унiверситет iменi Тараса Шевченка,03127, м. Київ, проспект академiка Глушкова 6,Українаe-mail: [email protected], [email protected]
54
Про зближення розподiлiв пуассонiвськогота числа хибних розв’язкiв системи
нелiнiйних випадкових рiвнянь у полi GF (2)В. I. Масол та М. В. Слободян, Україна
Розглянемо систему рiвнянь
gi(n)Xk=1
X1≤j1<· · ·<jk≤n
a(i)j1 · · · jkxj1 · · · xjk = bi , i = 1, 2, . . . , N, (1)
у полi GF (2) при умовi (А):коефiцiєнти a
(i)j1 · · · jk (1 ≤ j1 < · · · < jk ≤ n, k = 1, 2, . . . , gi(n) ,
i = 1, 2, . . . , N ) – незалежнi випадковi величини, Pa(i)j1 · · · jk = 1
= 1− Pa(i)j1 · · · jk = 0 = pik ;
елементи bi (i = 1, 2, ... , N) - результат пiдстановки в лiву частинусистеми (1) фiксованого n-вимiрного (0, 1)-вектора x0 , який має ρ(n)ненульових компонент;
функцiя gi(n) - невипадкова, gi(n) ∈ 2, 3, ... , n (i = 1, 2, ... , N).Позначимо через νn число хибних розв’язкiв (1), тобто число розв’язкiв
системи (1), вiдмiнних вiд x0.В доповiдi розглядаються оцiнки швидкостi збiжностi до граничного
розподiлу числа хибних розв’язкiв системи нелiнiйних випадкових рiвняньу полi GF (2) i, зокрема, пропонується наступний результат.
Теорема. Нехай виконуються умови: (А);
n−N = m, m = const, −∞ < m <∞ при n→∞;
δi1(n) ≤ pi1 ≤ 1− δi1(n), i = 1, N ;
для деякої функцiї ϕ(n)PNi=1 exp−εϕ(n)δi1(n) → 0, n→∞, ε = const, ε ∈ (0, 1);
для довiльного i = 1, 2, ... , N iснує таке t , що при n → ∞
pit =1
2, t ∈ 2, ..., gi(n).
Тодi для фiксованого k = 0, 1, 2, . . . при n→∞˛Pνn = k − λk
k!e−λ
˛≤ 4
„4eλ
log2 µ(n)
« 12 log2 µ(n)
,
де λ = 2m , µ(n) = nϕ(n) lnn
, µ(n) →∞ (n→∞) .
Кафедра теорiї ймовiрностей та математичної статистики,механiко-математичний факультет,Київський нацiональний унiверситет iменi Тараса Шевченка,проспект Глушкова, 6, Київ 03127, Українаe-mail: [email protected], [email protected]
55
Про асимптотичну нормальнiсть числахибних розв’язкiв системи нелiнiйних
випадкових рiвнянь в полi GF(2)
Володимир Масол, Свiтлана Слободян, Україна
Розглянемо над полем GF(2), що складається з двох елементiв, системурiвнянь
gi(n)Xk=1
X1≤j1<...<jk≤n
a(i)j1...jk
xj1 . . . xjk = bi (i = 1, . . . , N), (1)
яка задовольняє умову (А).Умова (А):1) Коефiцiєнти a
(i)j1...jk
(1 ≤ j1 < ... < jk ≤ n, k = 1, ..., gi(n), i = 1, ..., N)є незалежними випадковими величинами, якi приймають значення 1з ймовiрнiстю P
na(i)j1...jk
= 1o
= pik i значення 0 з ймовiрнiстю
Pna(i)j1...jk
= 0o
= 1− pik .2) Елементи bi (i = 1, ..., N) являють собою результат пiдстановки в лiву
частину системи (1) фiксованого n - вимiрного вектора x0 , в якому ρ (n)/n− ρ (n)/ компонент, що дорiвнюють одиницi /нулю/.
3) Функцiя gi(n), i = 1, ..., N – невипадкова,gi(n) ∈ 2, ..., n (i = 1, ..., N).
Через νn позначимо число хибних розв’язкiв системи (1), тобто числорозв’язкiв системи (1), вiдмiнних вiд вектора x0 .
Покладемо m = n − N , m = [log2 λ] , деλ = 1
2(1+α+ω)log2
ρ(n)ϕ(n) lnn
, ϕ(n) > 0 при n → ∞ , α > exp1 + 1α ,
ω > 0 .
Теорема 1 Нехай виконуються умови (А), при n→∞
λ→∞,
ω√λ→∞;
для довiльного i (i = 1, ..., N) iснує непорожня множина Ti така, що приn→∞
Ti ⊆ 2, 3, ..., gi(n) , Ti 6= ø,
δit(n) ≤ pit ≤ 1− δit(n), t ∈ Ti,
limn→∞
λB (ρ(n)− 1; 1) <∞,
56
де B (X,Y ) =NPi=1
exp
(−2
Pt∈Ti
δit(n)Ct−YX
);
(2 + (1 + α+ ω) ln 2)λ− lnλ
2+ lnB (εϕ(n), 0) → −∞ (n→∞),
де ε = const, 0 < ε < 1;
limn→∞
(− lnN + lnB (εϕ(n), 1)) < 0 (n→∞).
Тодi функцiя розподiлу випадкової величини νn−λ√λ
прямує до стандартноїнормальної функцiї розподiлу.
Доведення теореми базується на використаннi наступної леми.Лема. Нехай X та Y – випадковi величини, якi набувають цiлихневiд’ємних значень i λ∗ = MX . Якщо розподiли цих випадкових величинзмiнюються так, що
sup1≤r≤(1+α+ω)λ∗
˛M(X)r(M(Y )r)
−1 − 1˛ e2λ∗√
λ∗→ 0
i при всiх r ≤ (1 +α+ω)λ∗ M(Y )r ≤ C (λ∗)r для деякої сталої C , де M(ξ)rпозначає r–й факторiальний момент випадкової величини ξ, r ≥ 1 , то
max1≤t≤(1+ω)λ∗
|P X ≥ t − P Y ≥ t| → 0.
Зауваження. Можна показати, що в умовах теореми математичнесподiвання випадкової величини νn допускає представлення
Mνn = 2m (1 + γ(n))
або
Mνn = λ 2−ε(n) (1 + γ(n)) ,
де 0 ≤ ε(n) < 1 i γ(n) → 0 при n→∞ .
[В. I. Масол]Кафедра теорiї ймовiрностей та математичної статистики,механiко-математичний факультет,Київський нацiональний унiверситет iменi Тараса Шевченка,проспект Академiка Глушкова, 6, Київ, 03127, Українаe-mail: [email protected]
[С. Я. Слободян]Кафедра теорiї ймовiрностей i математичної статистики,механiко-математичний факультет,Київський нацiональнiй унiверситет iменi Тараса Шевченка,проспект Академiка Глушкова, 6, Київ, 03127, Українаe-mail: [email protected]
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Стратегiї запiзнення в сценарiях моделiрозподiленої ринкової економiки
Ю. П. Матусов, М. О. Заборовець, Україна
Дослiджуються деякi питання стiйкостi ринкової економiки, щосамостiйно розвивається, в умовах запiзнення змiнних у вiдповiдностi домоделi Магницького [1], яка є вiдмiнним прикладом розгляду хаосу внелiнiйному середовищi розподiленого ринку.
Нерiдко в середовищi ринку спостерiгаються хаотичнi змiни поведiнки,якi важко наперед передбачити та, здавалося б, стiйка система раптомруйнується пiд дiєю рiзких змiн своєї ж поведiнки.
Система ринкової економiки представлена моделлю з трьох нелiнiйнихдиференцiальних рiвнянь (1), два перших – описують змiну й дифузiюкапiталу, а третє описує попит в просторi технологiй пiд дiєю малогозапiзнення показника змiни норми прибутку.
Система рiвнянь (1) визначена з нормованими змiнними: для капiталуX , платоспроможного попиту Y та норми прибутку Z , якi описують змiнипоказникiв ринку в умовах малого запiзнення Z :8>><>>:
X ′(t) = b[(1− σ)Z(t− ε)− δ · Y (t)]X(t),Y ′(t) = [1− (1− δ)Y (t) + σ · Z(t− ε)]X(t),Z′(t) = a[Y (t)− d ·X(t)],ε ∈ (0, ε0), ε0 > 0,
(1)
де X(t) – нормована змiнна капiталу, Y (t) – нормована змiнна попиту,Z(t) – нормована змiнна норми прибутку, a , b , d – фiксованi параметри, щозалежать вiд дiяльностi в середовищi ринку, ε – запiзнення норми прибуткуна один часовий крок [2], σ , δ – параметри керування, якi визначаютьстабiльнiсть ринку (σ - характеризує “свободу” пiдприємцiв в прийняттiрiшень щодо розподiлення отриманого прибутку, δ – “тиск” з боку держави,який виявляється в податках, акцизах, митi тощо).
Нами було дослiджено, що мале запiзнення норми прибутку стабiлiзуєринок – формує атрактор Фейгенбаума (при δ=0,655), а в перспективiграничний цикл стягується в точку в результатi оберненої бiфуркацiїАндронова-Хопфа (при δ=0,687).
Лiтература
1. Магницкий Н. А. Математическая модель саморазвивающейсярыночной экономики. - Тр. ВНИИСИ АН СССР, 1991, - с.16-22.
2. Жданов Г. М. О приближенном решении систем дифференциальныхуравнений первого порядка с запаздывающим аргументом. УМН, 16,1(97), 2(98), 1961.- 143-148, 131-133.
Нацiональний Технiчний Унiверситет України"Київський Полiтехнiчний Iнститут", Київe-mail: [email protected]
58
Слабка збiжнiсть максимуму сумнезалежних випадкових процесiв
I.K. Mацак, Україна
Добре вiдоме значення дослiджень граничних розподiлiв максимумусум незалежних випадкових величин, наприклад, для теорiї масовогообслуговування чи теорiї страхування. Тому ця тематика iнтенсивновивчалася. Розглядався також i векторний випадок. Але для випадковихпроцесiв подiбнi питання здається не ставилися в лiтературi. В той жечас результати такого типу також представля- ють значний iнтерес длязастосувань.
Нехай X = X(s), s ∈ [0, 1] випадковий процес, EX(s) = 0 ,
(Xi) — послiдовнiсть незалежних копiй X , Sn(s) =nPi=1
Xi(s) ,
Sn(s) = max1≤k≤n Sk(s) .
Виявляється, що при широких умовах скiнченовимiрнi розподiливипадкового процесу Sn(s)/
√n збiгаються до скiнченовимiрних
розподiлiв процесу W (s) , де W (s) = sup0≤t≤1W (t, s) , а W (t, ·) деякийнескiнченовимiрний вiнеровський процес. У доповiдi ставиться задачадослiдження умов, при яких має мiсце слабка збiжнiсть
Sn(·)√n
D−→W (·) (1)
для функцiональних банахових граток.Зазначимо, що загальнi спiввiдношення (1) у просторах Lp чи C можна
застосувати до асимптотичного аналiзу деяких статистик типу Колмогорова-Смiрнова та ω2 .
Київський нацiональний унiверситетiменi Тараса Шевченка,03022, Київ - 22, просп. Глушкова, 2, корпус 6,факультет кiбернетикиe-mail: [email protected]
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Iснування та єдинiсть розв’язкустохастичного диференцiального рiвняння,що мiстить двопараметричний дробовий
броунiвський рух на площинi
Ю. С. Мiшура та C. A. Iльченко, Україна
Нехай (Ω,F , P ) — повний ймовiрнiсний простiр, на якому заданодвопараметричний дробовий броунiвський рух BHt : [0, T ]× Ω → R .
Розглянуто стохастичне диференцiальне рiвняння вигляду
Xt = X0 +
Z[0,t]
b(s,Xs)ds+
Z[0,t]
σ(s,Xs)dBHs = X0 + F
(b)t (X) +G
(σ)t (X), (1)
де t ∈ [0, T ] ⊂ R2+ , точку T фiксовано, BHt — дробове броунiвське поле з
параметрами Хюрста Hi ∈„
1
2, 1
«, i = 1, 2 , σ, b : [0, T ] × R → R — вимiрнi
функцiї, що задовольняють наступнi умови:
1) функцiя b(s, x) неперервна по s i ∀N > 0∃CN > 0 таке,що∀ |x|, |y| 6 N, |b(t, x)− b(t, y)| 6 CN |x− y|, ∀ t ∈ [0, T ];
2)∃C > 0 таке, що |b(t, x)| 6 C, |σ(t, x)| 6 C, t ∈ [0, T ], x ∈ R;
3)σ ∈ C3[0, T ]× C5(R);
4)∃C > 0 таке,що |Dσ(t, x)| 6 C, де D означає всiдиференцiювання, якi можна виконати згiдно з умовою 3).
9>>>>>>>>>=>>>>>>>>>;(2)
Пiд розв’язком рiвняння (1) будемо розумiти випадкове полеXt = Xt(ω) : [0, T ] × Ω → R , яке при м.в. ω ∈ Ω перетворює (1) натотожнiсть, а iнтеграл G
(σ)t (X) iснує для м.в. ω ∈ Ω як двопараметричний
узагальнений iнтеграл Лебега-Стiлтьєса.
Теорема 1 Нехай коефiцiєнти рiвняння (1) задовольняють умови 1) —4) з (2). Тодi рiвняння (1) має єдиний в класi Wα1,α2
1 розв’язок Xtна прямокутнику [0, T ] , причому для м.в. ω ∈ Ω , X ∈ H1−α1,1−α2 ,
∀ 1−Hi < αi <1
2, i = 1, 2 .
Зауваження 1 В статтях [1], [2] одержано оцiнки для норм у просторахтипу Бєсова узагальнених двопараметричних iнтегралiв вiдносно дробовихброунiвських полiв на площинi, спираючись на якi, вдалося встановитиумови iснування та єдиностi розв’язку стохастичного диференцiальногорiвняння (1).
60
Зауваження 2 В однопараметричному випадку теорему iснування таєдиностi розв’язку стохастичного диференцiального рiвняння, що мiститьдробовий броунiвський рух, доведено в статтi [3].
Лiтература
[1] Mishura, Yu. S. and Ilchenko, S. A. Some estimates for two-parameter gener-alized stochastic Lebesgue-Stieltjes integrals, Theory of Stochastic Processec.,Vol. 9(25), no. 3-4 (2003), pp. 87–100.
[2] Iльченко, С. А. Оцiнки дробових норм вiд iнтегралiв по дробовомуброунiвському полю, Доповiдi НАН України, no. 4 (2005), pp. 12–17.
[3] Nualart, D.; Rascanu, A. Differential equations driven by fractional Brownianmotion, Collect. Math. 53, no. 1 (2002), 55–81.
Київський нацiональний унiверситет iменi Тараса Шевченка,64 Володимирська, 01033 Київ, Українаe-mail: [email protected]
Черкаський нацiональний унiверситет iменi Богдана Хмельницького,81 б-р. Т. Шевченка, 18031 Черкаси, Українаe-mail: [email protected]
61
Поняття обмеженого арбiтражу длябагатоперiодного ринку з дискретним часом
Мiшура Ю.С., Шеляженко П.С., УкраїнаШироко застосованим методом моделювання фiнансового ринку є
наступний: iнвестор працює на ринку з випадковими цiнами активiв, i за часT перетворює початковий капiтал V0 в деяку кiнцеву кiлькiсть VT
Одним з найбiльш цiкавих питань про такi ринки є питання проiснування арбiтражної можливостi, тобто чи може iнвестор отримати деякийприбуток без ризику. Багато праць присвячено цьому питанню. Для моделейз дискретним часом вiдсутнiсть арбiтражної можливостi еквiвалентнаiснуванню еквiвалентної мартингальної мiри для цiнового процесу. Цей фактшироко вiдомий, вiн пояснюється детально скажiмо у [1] та [2]. Додатковаiнформацiя про класичне поняття арбiтражу для ринкiв з дискретним танеперервним часом може бути знайденим скажiмо у [3].
Для великих фiнансових ринкiв ми припускаємо необмеженi кiлькостiбудь-якого активу на ринку. З цього випливає що арбiтражна можливiстьозначає для iнвестора можливiсть отримати будь-який конкретний прибутокбез ризику. У дiйсностi ситуацiя трохи iнша: арбiтражнi можливостi насправдiiснують на деяких ринках, але iнвестор може використовувати їх лише вобмеженiй мiрi, тому що ринок обмежений сам по собi (на ньому лишескiнченна кiлькiсть активiв). Така ситуацiя i є предметом розгляду даноїстаттi, ми називаємо її обмеженим або ε -арбiтражем. Вивчення iснування ε -арбiтражу вимагає iншого математичного апарату, нiж той що застосовуєтьсядля звичайного арбiтражу, бо ринок без ε -арбiтражної можливостi можебути арбiтражним у звичайному сенсi. Поняття ε -арбiтражу було впершерозглянуто у [4]. В цiй статтi ми продовжуємо вивчення ε -арбiтражу. Теорiя,аналогiчна теорiї звичайного арбiтражу, будується i для ε -арбiтражу. В цiйстаттi розглядається перша фундаментальна теорема фiнансової математикидля моделi з випадковими початковими даними, а також дослiджуєтьсябагатоперiодна модель. Теореми аналогiчнi класичнiй теоремi Даланга-Мортона-Вiллiнджера ([2]) доведенi для багатоперiодного фiнансового ринку.
Отже, наведемо використанi означення i припущення. Нехай(Ω, F, P ) це повний ймовiрнiсний простi, оснашений фiльтрацiєю(Ft)t=0,T , FT = F . Нехай S = (St) це Ft -адаптований d + 1 -вимiрнийпроцес (вектор цiн). Стратегiєю будемо називати (d + 1) -вимiрний процесξ = (ξ0, ξ) = (ξ0t , ξ
1t , ..., ξ
dt ) , який є адаптованим до Ft .дисконтований цiновий
процес через Xit , а дисконтованим процесом капiталу, асоцiйованим
зi стратегiєю ξ , будемо називати процес, заданий наступним чином:V0 := ξ0 ·X0; Vt := ξt−1 ·Xt, t = 1, T .Означення 1. Багатоперiодний фiнансовий ринок допускає ε -арбiтраж,якщо iснує самофiнансована стратегiя ξ = (ξt) , що є обмеженою в сенсi
‖ξt‖1 :=
‚‚‚‚‚dXi=1
|ξit|
‚‚‚‚‚L∞(Ω)
≤ 1, ∀t = 0, T − 1 (1)
62
асоцiйований з якою процес капiталу задовольняє умови: V0 ≤ −ε ,PVT ≥ 0 = 1 , PVT > 0 > 0.
Це означення багато у чому нагадує класичне означення арбiтражу. Алеможно помiтити двi важливi вiдмiнностi. По-перше, умова 1 означає що сумамоделей компонент нашої стратегiї не може перевищувати 1. Ще робитьнашу стратегiю обмеженою. По-друге, аби мати ε -арбiтражну можливiстьми маємо зробити "стрибок" вiд початкового капiталу −ε < 0 до кiнцевогокапiталу бiльшого за нуль. Для класичного арбiтражу умова виглядала як:V0 ≤ 0, PVT ≥ 0 = 1, PVT > 0 > 0.Тепер можемо сформулювати наступну теорему для одноперiодного ринку:Теорема 1. Для нашого ринку наступнi умови є еквiвалентними:1. Фiнансовий ринок є ε -безарбiтражним;2. Iснує мiра P ∗ еквiвалентна до P , така що dP∗
dP≤ C для деякого C > 0 ,
i EP∗ [ maxi=1,d
|EP∗ [Xi1 −Xi
0/F0]| ] ≤ ε .
Неважко помiтити аналогiї з першою фундаментальною теоремоюфiнансової математики, якщо взяти ε = 0 .
Для багатоперiодного ринку нам необхiднi додатковi означення. Нехай
RεT := VT : VT =TPt=1
ξt · 4Xt − ε , де ξ це довiльна самофiнансована
стратегiя, яка обмежена у сенсi 1, 4Xt := Xt − Xt−1 . AεT := RεT − L0+
= V − V + : V ∈ RεT , V+ ∈ L0
+ . Ця множина мiстить усi зобов’язання,якi можуть бути хеджованi деякою стратегiєю з початковим капiталом −ε .Теорема 2. Наступнi умови еквiвалентнi:1. RεT ∩ L0
+ ⊂ 0 (ε -безарбiтражнiсть)2. AεT ∩ L0
+ ⊂ 03. AεT ∩ L0
+ ⊂ 0 i AεT = AεT4. AεT ∩ L
0+ ⊂ 0
5. ∃P ∗ ∼ P , така що EP∗
»TPt=1
maxi=1,d
| EP∗ [4Xti /Ft−1] |
–≤ ε ,
де 4Xt := Xt − Xt−1 , причому похiдна Радона-Нiкодима є обмеженою,тобто dP∗
dP≤ C , для деякого C > 0 .
[1] Harrison M., Pliska S, Martingales and stochastic integrals in the theory of continuoustrading, Stochastic Process. Appl., 1981, Vol. 11, p. 215-260
[2] Dalang R.C., Morton A., Willinger W., Equivalent martingale measures and no-arbitrage in stochastic securities market models, "Stochastics and Stochastics Re-ports", 1990, Vol. 29, p. 185-201
[3] Follmer H., Schied A., Stochastic finance. An introduction in discrete time. 2ndedition., "Walter de Gruyter", 2004, 459 p.
[4] Мiшура Ю.С., Основна теорема фiнансової математики для обмеженогоарбiтражу, “Прикладна статистика. Актуарна та фiнансова математика”, 2003,1-2, стр. 49-54
[5] Kabanov Yu. M., Arbitrage theory, Manuscript, 2000, Laboratoire de Mathematiquesde Besancon
63
До оцiнок розв’язкiв стохастичнихдиференцiальних рiвнянь в гiльбертових
просторах
Наконечний Олександр Григорович, Україна
Дослiджуються задачi про оцiнку розв’язку стохастичних диференцiйнихрiвнянь в гiльбертовому просторi H вигляду
dx(t) = A(t)x(t) +B(t, x(t))dW1(t),
x(t0) = Cx0
за спостереженнями функцiї y(t), 0 ≤ t ≤ T, iз гiльбертового простору H1,що задовольняє рiвнянню
dy(t) = H(t)x(t)dt+ σ(t, x(t))dW2(t),
де A(t), B(t, x), C,H(t), σ(t, x) – лiнiйнi обмеженi оператори, W1(t),W2(t) –стандартнi незалежнi вiнеровi процеси в деяких гiльбертових просторах, x0
– невiдомий вектор iз множини G гiльбертового простору H0.Мiнiмаксна лiнiйна середньоквадратична оцiнка визначається iз умови
infl∈L
supx0∈G
M |x(T )− l(y(·))|2 = supx0∈G
M |x(T )− l(y(·))|2,
де L – множина лiнiйних оцiнок.Вивчаються умови iснування таких оцiнок, умови їх незмiщеностi, а
при певних обмеженнях показано, що такi оцiнки задовольняють деякимдиференцiйним рiвнянням.
Лiтература
[1] Наконечний О.Г. Оцiнювання параметрiв в умовах невизначеностi//Науковi записки КНУ. – Т.VII., 2004. – C.102–112.
Київський нацiональний унiверситет iменi Тараса Шевченка,факультет кiбернетики,03127, Київ, просп. Академiка Глушкова, 2, корп.6e-mail: [email protected]
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Узагальнена Γ-конструкцiя
Нiкiфоров Роман, Україна
Розглядається узагальнена конструкцiя побудови мультифрактальноїмножини канторiвського типу запропонованої Я. Гончаренко. На базiузагальненої Γ–конструкцiї будуються сингулярно неперервнi iмовiрнiснiмiри, для яких мультифрактальнi характеристичнi мiри першого порядкубудуть сингулярними.
Лiтература
[1] Торбiн Г. М. Мультифрактальний аналiз сингулярно неперервнихiмовiрнiсних мiр // Укр. мат. журн. — 2005. —57, 5. — C. 706–721.
[2] Гончаренко Я. В. Мультифрактальнi множини i розподiли ймовiрностей.// Науковi записки НПУ iменi М.П.Драгоманова. Фiзико-математичнiнауки — 1999. —1. — С. 228 – 233.
Нацiональний педагогiчний унiверситет iменi М. П. Драгоманова, Київe-mail: [email protected]
65
Оцiнка точностi моделювання субгауссовихполiв
Пашко Анатолiй, Україна
В роботi вивчаються оцiнки точностi моделювання субгауссових полiв.Нехай X = X(t), t ∈ T - строго субгауссове випадкове поле. Позначимо
ξ(u) =r`N`σ(−1)(u)
´´`lnN
`σ(−1)(u)
´´1/2де
supρ(t,s)≤h
`E`X(t)−X(s)
´2´1/2 ≤ σ(h)
i N(ε) = NT (ε) - число точок в мiнiмальному покриттi деякого простору`T, ρ
´замкненими кулями радiусу ε > 0 . Функцiя r =
˘r(t), t ≥ 1
¯така, що
r(t) ≥ 0, i r(1) = 0 .Теорема. Якщо для функцiї r =
˘r(t), t ≥ 1
¯такої що r(t) ≥ 0 , r(1) = 0 ,
i для опуклої функцiї s(u) = r`exp˘u¯´
виконується умоваZ0+
ξ(u)du <∞,
то для ε > 0 , 0 < p < 1 має мiсце оцiнка
P
supt∈T
‖X(t)‖ > ε
ff≤ 2 inf
λ>0exp
λ2
2Ak(p)− λx
ffLk(λ, p),
де
A1(p) =1
1− p
„γ2w +
z2
p(1− p)
«,
i
L1(λ, p) =
„r(−1)
„λI(zp)√2p(1− p)
««2
та w - довiльна точка з T , а z = σ`supt∈T ρ(t, w)
´, γw =
`E‖X(w)‖2
´1/2 ,
A2(p) =1
1− p
„θ20 +
pβ2
1− p
«,
L2(λ, p) =
„r(−1)
„λ√2
„θ0ξ(pβ) +
I(p2β)
p(1− p)
«««2
,
iθ0 = supt∈T
`E‖X(t)‖2
´1/2 , β ≤ σ`infs∈T supt∈T ρ(t, s)
´.
ПВНЗ "Європейський унiверситет", Київ, Українаe-mail: [email protected]
66
Про один метод моделювання субгауссовихполiв, що зображуються у виглядi кратних
рядiвПашко Анатолiй, Зелепугiна Iрина, Україна
В роботi вивчаються оцiнки точностi моделювання субгауссових полiв, щозображуються у виглядi кратних рядiв у рiзних функцiональних просторах.Випадкове поле
Xn1 (t) =
nXm=1
Xk1+k2+...+kd=m
ξkφk(t)
розглядається як модель випадкового поля
X(t) =
∞Xm=1
Xk1+k2+...+kd=m
ξkφk(t)
де ξk – незалежнi сторого субгауссовi випадковi величини, φk(t) –функцiї з деякого функцiонального простору.
Теорема 1. Якщо iснує неспадна послiдовнiсть bk i θ ∈ (0, 1) , така, щомають мiсце умови
∞Xm=1
Pm1 δm`1/bm
´| lnβm
`θ´|1/2 <∞
nXm=l+1
Pml+1δm`1/bm
´| lnβm
`θ´|1/2 → 0 при n→∞
та Xk1+k2+...+kd=n
Pnl+1 lnβn(θ)|1/2
bn→ 0 при n→∞
то P ||c(t)(X(t)−Xn1 (t))||C > ε → 0 при n → ∞ i для ε > Dn/(1 − θ) має
мiсце
P||c(t)(X(t)−Xn1 (t))||C > ε ≤
ZRd
|c(t)|dµ(t) exp
8><>:−“ε− Dn
1−θ
”2
(1− θ)2
2A2n
9>=>; ,
де
Dn = 21/2∞X
m=n+1
Pm1 δm(1/bm)| lnβm(θ)|1/2, An =
∞Xm=n+1
Pm1 δm(1/bm)
В роботi розглянуто деякi конкретнi приклади моделювання субгауссовихвипадкових полiв.
ПВНЗ "Європейський унiверситет",Київський державний унiверситет технологiй та дизайнуe-mail: [email protected]
67
Про моделювання процесiв Кокса керованихвипадковим логарифмiчно гауссовим полем
Погорiляк Олександр Олександрович, Україна
Нехай Ω,F,P – стандартний ймовiрнiсний простiр, B–σ -алгебраборелiвських множин T , T ⊂ Rn , Y (t) , t ∈ T – однорiдне,центроване, гауссове, неперервне в середньому квадратичному випадковеполе, EY (t) = 0 , B (t− s) = EY (t)Y (s) .
Означення 1 Логгауссовим процесом Кокса або процесом Коксакерованим логгауссовим випадковим полем exp Y (t) називаєтьсяпроцес ν (B) , B ∈ B визначений у такий спосiб:
1) якщо B1∩B2 = ∅ , B1, B2 ∈ B , то випадковi величини ν (B1) та ν (B2)незалежнi;
2) P ν (B) = k / Y (t) , t ∈ T = exp−µ(B) (µ(B))k
k!, k = 0, 1, 2, . . . ,
µ (B) =
ZB
exp Y (t, ·) dt,
Y (t, ·) , t ∈ T — реалiзацiя процесу Y (t) , t ∈ T .
Оскiльки ν (B) , B ∈ B це подвiйно стохастичний випадковий процес, тойого модель будується в два етапи. Спочатку моделюється гауссове випадковеполе Y (t) , t ∈ T , далi розглядається розбиття DT областi T = [0, T ]n
Bi1,...,in =nhti11 , t
i1+11
”× . . .×
htinn , t
in+1n
” ˛timm < tim+1
m ,
tim+1m − timm = d =
T
k, k ∈ N, m = 1, n, im = 0, k − 1
ff,
i на кожному елементi Bi1,...,in будується модель пуассонiвської випадковоївеличини з вiдповiдним середнiм.
Позначимо eY (t) – модель поля Y (t) ,eµ (Bi1,...,in) =RBi1,...,in
expneY (t)
odt , eν (Bi1,...,in) – модель ν (Bi1,...,in) ,
тобто модель пуассонiвської випадкової величини з середнiм eµ (Bi1,...,in) .Зрозумiло, що модель логгауссового процесу Кокса
ν (B) , B ∈ B можна вважати допустимою, якщо умовнiймовiрностi pkY (Bi1,...,in) = P ν (Bi1,...,in) = = k / Y (t) , t ∈ T таepkY (Bi1,...,in) = P
n eν (Bi1,...,in) = k / eY (t) , t ∈ To
вiдрiзняються мало, атакож ймовiрнiсть того, що число точок ν (Bi1,...,in) (вiдповiдно i eν (Bi1,...,in))буде бiльше одиницi, буде також мала. Якщо в елемент розбиття Bi1,...,inпопадає одна точка, то розмiщуємо її в центрi, якщо двi i бiльше, торозмiщуватимемо їх в Bi1,...,in довiльним чином.
Нехай Rn+,U ,Φ – вимiрний простiр, де U – борелiвська σ - алгебра
множин, Φ – скiнченна мiра За модель процесу Y (t) приймаємо суму виду
68
eY (t) =
N−1Xi1,...,in=0
cos“t,λ
“λi11 , . . . , λ
inn
””Z1 (∆ (i1, . . . , in))
+
N−1Xi1,...,in=0
sin“t,λ
“λi11 , . . . , λ
inn
””Z2 (∆ (i1, . . . , in)),
де λ`λi11 , . . . , λ
inn
´точки розбиття DΛn :
∆ (i1, . . . , in) =nhλi11 , λ
i1+11
”× . . .×
hλinn , λ
in+1n
” ˛λimm < λim+1
m ,
λim+1m − λimm =
Λ
N, Λ ∈ R+, N ∈ N, m = 1, n, im = 1, N − 1
ff,
Z1 (S) та Z2 (S) , S ∈ U , некорельованi випадковi мiри пiдпорядкованi мiрiΦ .
Означення 2 Скажемо, що модель логарифмiчно гауссового процесу Коксаν (Bi1,...,in) , Bi1,...,in ⊂ B наближає його з точнiстю α, 0 < α < 1 танадiйнiстю 1− γ, 0 < γ < 1 , якщо виконується нерiвнiсть
P
max
Bi1,...,in∈B| pkY (Bi1,...,in)− epkY (Bi1,...,in) | > α
ff< γ.
Теорема 1 Нехай Y (t) – однорiдне, центроване, неперервне в середньомуквадратичному випадкове поле, тодi модель логгауссового процесу Коксаν (Bi1,...,in) наближає його з точнiстю α та надiйнiстю 1 − γ , якщовиконуються умови:
MN < exp −v2B (0) ,
2 kn„− lnMN
v2B (0)
«− ln MN2v2B(0)
exp
− ln2MN
2v2B (0)
ff< γ,
де
MN =dn (v1 + 1)
12 A
12N
α,
AN = 22−2an2a d2aΛ2a
N2aΦ (Λn) +B (0)− Φ (Λn) ,
B (τ ) – коварiацiйна функцiя поля Y (t) , v2 = v1v1−1
, v1 – будь-яке додатнедiйсне число, a ∈ [0, 1] .
Кафедра теорiї ймовiрностей та математичної статистики,механiко-математичний факультет,Київський нацiональний унiверситет iм. Тараса Шевченкаe-mail: alex
69
Правильна змiна у гiллястому випадковомублуканнi
Полоцький Сергiй Вiкторович, Україна
Розглядається гiллясте випадкове блукання з нескiнченним числом нащадкiвта пов’язаний з ним невiд’ємний мартингал Wn
Wn :=X|u|=n
eγAu
[EP|u|=1 e
γAu ]n=X|u|=n
Yu,
де Au, |u| = n позначає положення на дiйснiй вiсi особи u n−го поколiння,γ > 0 таке, що E
P|u|=1 e
γAu <∞. Нехай dk−мартингал-рiзниць, тобто
Wn = 1 +
nXk=1
dk, n = 1, 2, . . .
Зауважимо, що W0 = EWn = 1. Введемо також позначення
S :=
1 +
∞Xk=1
d2k
!1/2
, W ∗ := supn≥0
Wn , ∆ := supn≥1
|dn|.
Так як Wn невiд’ємний мартингал, то вiн збiгається м.н. до випадковоївеличини W . Тому введемо ще одне позначення
M := supn≥0
|W −Wn|.
Виконується таке твердження
Теорема 1 Припустимо, що iснують β > 1 та ε > 0 такi, що
kβ := EX|u|=1
Y βu < 1 , EX|u|=1
Y β+εu <∞ i (1)
PW1 > x ∼ x−βL(x),
де запис f(x) ∼ g(x) означає, що limx→∞f(x)
g(x)= 1, а L(x) позначає функцiю,
що правильно змiнюється на нескiнченностi. Тодi
I. PrW ∗ > x ∼ Pr∆ > x ∼ PrS > x ∼ 1
1− kβPrW1 > x;
II. PrM > x ∼ PrW > x ∼ 1
1− kβPrW1 > x,
Умови (1) теореми 1 гарантують, що мартингал Wn буде рiвномiрноiнтегровним. Тому гранична величина W буде додатною.
КНУ iм. Шевченка, факультет кiбернетикиe-mail: [email protected]
70
Методична робота кафедри теорiїймовiрностей КНУ iз статистики, стохастики
та їх застосуваньОлександр Пономаренко, Україна
Протягом останнiх десяти рокiв кафедра теорiї ймовiрностей таматематичної статистики Київського нацiонального унiверситету iменiТараса Шевченка проводила iнтенсивну методологiчну роботу в галузяхвикладання статистики, стохастики та їх застосувань, пов’язану iзвведенням нової математичної спецiальностi "Статистика" з вузькимиспецiалiзацiями "Фiнансова та актуарна математика" i "Математичнаекономiка та економетрика" на механiко-математичному факультетi КНУ таiз вдосконаленням традицiйної спецiалiзацiї кафедри "Теорiя ймовiрностей таматематична статистика".
За вказаний перiод часу було розроблено чотири варiанти навчальнихпланiв спецiальностi "Статистика" в 1996,1998,2002 та 2006 роках, виданочотири збiрники навчальних програм нормативних та спецiальних курсiвдля рiвнiв бакалаврiв, спецiалiстiв та магiстрiв. Кафедра виконала рядмасштабних мiжнародних освiтнiх проектiв з розвитку та удосконаленнястатистичної освiти на Українi в рамках освiтньої програми TEMPUS-TACIS Європейського Союзу. Було проведено вiсiм мiжнародних науково-методичних шкiл зi спецiалiзацiй кафедри, видано серiю збiрок праць та тезвiдповiдних форумiв для широкого користування в iнших вузах України.
Значну увагу кафедра присвятила пiдготовцi та виданню пiдручникiв тапосiбникiв для вказаних трьох спецiалiзацiй. Загалом за останнi одинадцятьрокiв було видано понад тридцять назв книжок з таких дисциплiн якматематична економiка, мiкроекономiчний аналiз, макроекономiчнийаналiз, системний аналiз в економiцi та менеджментi (О.I.Пономаренко,М.О. Перестюк, В.М.Бурим), фiнансова математика (Ю.С.Мiшура,О.I.Пономаренко,О.Д.Борисенко), актуарна математика (М.Й.Ядренко,О.I.Пономаренко, А.Я.Оленко), економетрика, регресiйний аналiз такомп’ютерна статистика (М.М.Леоненко, Р.Є.Майборода), дискретнаматематика (М.Й.Ядренко), вейвлет аналiз (Ю.В.Козаченко), теорiяекстремальних задач та опуклий аналiз (М.П.Моклячук), теорiя ймовiрностейта математична статистика (М.В.Карташов) та ряду iнших дисциплiн.
Зараз кафедра проводить значну методологiчну та органiзацiйну роботупо пiдготовцi та впровадженню системи пiслядипломної освiти для пiдготовкисертифiкованих актуарiїв та фiнансових аналiтикiв для України, щовiдповiдає мiжнародним стандартам та вимогам i використовує досвiдлiдуючих в цих галузях країн ЄС. З цiєю метою розробляється системапрофесiйних навчальних курсiв та посiбникiв. Цей напрямок роботи кафедрипiдтримується проектом TEMPUS PROJECT IB - JEP - 25054-2004.
Київський нацiональний унiверситет iменi Тараса Шевченка,Механiко-математичний факультет, e-mail:[email protected]
71
Метрична, ймовiрнiсна та фрактальна теорiїϕ∞-розкладiв дiйсних чисел
М.В. Працьовитий, Україна
Нехай Q∞ = qi , i ∈ N — нескiнченний набiр дiйсних чисел, для якихмають мiсце властивостi:
1) qi > 0, ∀j ∈ N ; 2)
∞Xi=1
qi = 1.
Легко довести, що для кожного числа x ∈ [0; 1] iснує послiдовнiстьαk ∈ N , αk = αk(x) , така, що
x = βα1 +
∞Xk=1
"βαk
k−1Yi=1
qαi
#, (1)
де β0 = 0 , βm =m−1Pj=1
qj . Вираз (1) числа x називається його Q∞ -
представленням. Множина
∆c1...cm =
"βc1 +
mXk=1
βck
k−1Yi=1
qci ; βc1 +
mXk=1
βck
k−1Yi=1
qci + βcm+1
m−1Yj=1
qcj
#називається цилiндричною множиною (цилiндром) рангу m з основоюc1c2...cm .
Нехай ϕ(x) — неперевна функцiя розподiлу на вiдрiзку [0; 1] ,
∆ϕc1...cm
= ϕ(∆c1...cm).
Тодi подання числа u у виглядi
u = ∆ϕα1...αk... =
∞\k=1
∆ϕα1...αk
називається ϕ∞ -розкладом.В доповiдi пропонуються основи метричної, ймовiрнiсної та фрактальної
теорiй ϕ∞ -розкладiв дiйсних чисел.
Лiтература
[1] Працьовитий М.В. Фрактальний пiдхiд у дослiдженнях сингулярнихрозподiлiв. — Київ: Вид-во НПУ iменi М.П.Драгоманова, 1998. — 296 с.
[2] Турбин А.Ф., Працевитый М.В. Фрактальные множества, функции,распределения. — Киев: Наукова думка, 1992. — 208с.
НПУ iменi М.П.Драгоманова, Україна, Київ
72
Фрактальнi властивостi розподiлiв елементiвтопологiчних просторiв, заданих
розподiлами своїх f -символiв
Працьовитий М.В., Фещенко О.Ю., Україна
Нехай s — фiксоване натуральне число, бiльше 1 ; A = 0, 1, ..., s − 1 —алфавiт; f — неперервна строго зростаюча функцiя на вiдрiзку [0; 1] ;
∆sc1c2...cm
=
"mXi=1
s−ici;1
sm+
mXi=1
s−ici
#—
цилiндрична множина рангу m з основою c1c2...cm , що вiдповiдає s -
адичному поданню чисел вiдрiзка [0; 1] , x = ∆sα1...αk... =
∞Tm=1
∆sα1...αm
,
∆fc1...cm
= f(∆sc1...cm
) , ∆fc1...cm... = f(∆s
c1...cm...) , ||qik|| — матриця ( i = 0,1, ..., s− 1 , k = 1, 2, ...), причому
1) qik > 0, 2) q0k + q1k + ...+ q(s−1)k = 1, ∀k ∈ N ;
|∆fc1...cmc|
|∆fc1...cm |
= ϕ(c1, ..., cm, c).
Дослiджується розподiл випадкової величини ξ
ξ = ∆fη1...ηk...,
визначений розподiлами f -символiв ηk , зокрема, коли1) ηk — незалежнi;2) утворюють ланцюг Маркова.Вивчаються фрактальнi властивостi мiнiмальної замкненої множини,
на якiй зосереджена ймовiрнiсть (топологiчного носiя), суттєвого носiящiльностi, а також розмiрнiсть Хаусдорфа-Безиковича вiдповiдноїймовiрнiсної мiри. Найбiльше уваги придiляється випадку, коли f —функцiя, що зберiгає фрактальну розмiрнiсть та коли f — сингулярнафункцiя.
Нехай B = A1, ..., Am , m = 2s−1 — впорядкована множина всiхнепорожних пiдмножин алфавiту A , ni(Vk, n) — число спiвпадань множинV1 , ... Vn (перших n з послiдовностi Vk з множиною A . Якщо iснує границя
limn→∞
ni(Vk, n)
n,
то її називатимемо частотою множини Ai в послiдовностi Vk iпозначатимемо νi(Vk) .
73
Для фiксованої послiдовностi Vk знайдено формулу для обчисленнярозмiрностi Хаусдорфа-Безиковича множини
M [τ0, ..., τs−1] = x : x = ∆fα1...αk..., αk ∈ Vk, νi(Vk) = τi.
Теорема 1 Якщо ϕ(c1, ..., ck, c) = qc = qck , Vn+j = Vn ∀j ∈ N , торозмiрнiсть Хаусдорфа-Безиковича α множини
C[Vk] = x : x = ∆fα1...αk..., αk ∈ Vk
задовольняє умову:nYi=1
Xa∈Vi
qαa = 1.
Теорема 2 Якщо всi стовпцi матрицi ||qjk|| однаковi, тобто qjk = qj ,i iснують частоти vi = vi(Vk) для всiх Ai , i ∈ 1, ..., 2s − 1 , торозмiрнiсть Хаусдорфа-Бiллiнгслi α множини
C[Vk] = u : u = ∆fα1...αk..., αk ∈ Vk ⊂ A
задовольняє умову2s−1Yi=1
0@Xa∈Ai
qαa
1Avi
= 1.
Множини цього типу для дослiджуваних випадкових величин виступаютьспектрами та носiями розподiлiв.
Лiтература
[1] Працьовитий М.В. Фрактальний пiдхiд у дослiдженнях сингулярнихрозподiлiв. — Київ: Вид-во НПУ iменi М.П.Драгоманова, 1998. — 296 с.
[2] Фещенко О.Ю. Про символьний спосiб задання функцiй i розподiлiвймовiрностей // Науковий часопис НПУ iменi М.П.Драгоманова. Серiя1. Фiзико-математичнi науки. — 2004, No 5. — С. 248-266.
[3] Працьовитий М.В., Фещенко О.Ю. Математичнi моделi двостороннiхдинамiчних конфлiктiв i Q -представлення чисел // Науковi запискиНПУ iменi М.П.Драгоманова. Фiзико-математичнi науки. — No 4, 2003.— С. 260-269.
НПУ iменi М.П.Драгоманова, Київ, Україна
74
Монiторинг банкiвської системи України:економетрична модель
Рязанцева В.В. (Україна)
Важливим iнструментом державного регулювання економiчнимипроцесами є кредити, тобто економiчнi вiдносини, якi виникають мiжсуб’єктами ринку з приводу перерозподiлу вартостi на засадах поверненостi,строковостi i платностi. Рушiйним мотивом такого перерозподiлу є отриманнядодаткового доходу кожним iз суб’єктiв кредитних вiдносин. Кредити, по-перше, дають змогу зосередити капiтал у найбiльш прiоритетних сферахекономiчної дiяльностi, здiйснитипереорiєнтацiю виробництва й стабiлiзувати економiку, по-друге, сприяютьекономiї витрат грошового обiгу; дозволяють прискорити обiг грошей,запроваджувати прогресивнi системи розрахункiв, по-третє, дають змогумiнiмiзувати кредитний ризик, реалiзувати цiльовий характер кредиту:стимулюють позичальника рацiонально й ефективно використовуватизапозиченi кошти.
За допомогою кредиту вiдбувається стимулювання економiчногозростання, забезпечується перелив капiталу у найефективнiшi сферигосподарського життя суспiльства, пiдвищується ефективнiсть грошовогообiгу. Вiн є важливим iнструментом державного регулювання економiчнимипроцесами.
Залежно вiд об’єкту кредитних вiдносин розрiзняють товарну та грошовуформи кредиту. За суб’єктами кредитних вiдносин видiляють такi видикредиту: комерцiйний, банкiвський, державний, мiжнародний,
Провiдним видом кредитування в ринковiй економiцi є банкiвськийкредит. Система банкiвського кредитування базується на використаннi рядубазових принципiв кредитування, а саме цiльового характеру кредитування,поверненостi кредиту, строковостi кредиту, платностi та забезпеченостiкредиту. Роль кредиту в економiцi виявляється наступним чином: завдякикредиту зменшується час на задоволення господарських та особистихпотреб; кредит виступає як опора сучасної економiки, невiд’ємний елементекономiчного розвитку; кредитори мають можливiсть отримати додатковiгрошовi кошти при передачi певної суми вiльних ресурсiв позичальнику.
На сьогоднiшнiй день великий бiзнес країни вже отримав альтернативубанкiвським кредитам у виглядi виходу на позичковi ринки, а також у виглядiпублiчного розмiщення акцiй, що стосується малого та середнього бiзнесу,тому основнi свої надiї вiн покладає на мiкрокредитування. Вже минуло вiсiмрокiв вiд заснування програми "Мiкрокредитування", i даний сегмент ринку внашiй країнi зростає великими темпами. Лiдером по данiй програмi виступаєвiдомий український банк ЗАТ КБ "ПриватБанк", оскiльки на сьогоднiшнiйдень вiн кредитує за цiєю програмою майже 50% попиту.
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Кредитний портфель ЗАТ КБ "ПриватБанку" по програмi"Мiкрокредитування" (CC) залежить вiд наступних факторiв: долi банку наринку мiкрокредитування в країнi (PB), кiлькостi профiнансованих проектiв(QP), кiлькостi клiєнтiв за даною програмою (QC), суми мiкрокредитiв поУкраїнi (AM), частки проблемних кредитiв в портфелi мiкрокредитування(PC), кiлькостi нових пiдприємств на Українi (QE).
На основi статистичної iнформацiї за чотири роки(джерело: www.privatbank.com.ua) побудовано регресiйну модель у виглядi:
CC = a0 + a1 · PB + a2 ·QP + a3 ·QC + a4 ·AM + a5 · PC + a6 ·QE + ζ
Проведено наступнi дослiдження:
• аналiз динамiки факторiв, визначено вектор прогнозних значень цихоб’єктiв на наступний перiод;
• перевiрка факторiв на мультиколiнеарнiсть за алгоритмом Феррара-Глоубера, виявлено зв’язки мiж QC i AM та мiж AM i QE;
• оцiнка невiдомих параметрiв моделi методом найменших квадратiв;
• обчислення вiдносної похибки та її стандартної помилки;
• перевiрка адекватностi моделi, статистичнi висновки;
• оцiнка коефiцiєнтiв кореляцiї та детермiнацiї
• перевiрка значущостi оцiнок параметрiв та вибiркового коефiцiєнтакореляцiї;
• оцiнка коефiцiєнта еластичностi
• дослiдження особливих випадкiв поведiнки стохастичної складової(автокореляцiя- не визначена за критерiєм Дарбiна-Уотсона,гетероскедастичнiсть - наявна за параметричним тестом ГольдфельдаКвандта);
• переоцiнка параметрiв моделi узагальненим методом найменшихквадратiв з метою урахування можливої автокореляцiї тагетероскедастичностi.
В результатi проведених дослiджень розрахованi зони надiйностi регресiї пристандартному рiвнi значущостi; обчислено прогнозне значення портфелюмiкрокредитування "ПриватБанку", яке становить 93,765 млн.дол.; верхнямежа iнтервалу надiйностi не перевищить значення 135,161 млн дол.;прогнозується, що в середньому значення кредитного портфелю неперевищить 98,127 млн дол.
Кафедра статистики та економетрiї,облiково-фiнансовий факультет, КНТЕУ,02156 Київ-156, вул.Кiото, 19e-mail: [email protected]
76
Iнтерполяцiя однорiдного та iзотропноговипадкового поля в центрi сфери за
рiвномiрно розподiленими спостереженнямина сферi
Наталiя Семеновська, Україна
Нехай ξ(x), x ∈ Rn , однорiдне та iзотропне випадкове поле на евклiдовомупросторi Rn, тобто, Eξ(x) = const (для простоти, не втрачаючи загальностi,будемо вважати Eξ(x) = 0), E|ξ(x)|2 < ∞ i кореляцiйна функцiяϕ(|x − y|) = Eξ(x)ξ(y) залежить лише вiд вiдстанi |x − y| мiж точками xта y.
Зафiксуємо на сферi Sn радiусу r деяку скiнченну множину спостереженьX = x1, . . . , xN . Нехай xk = (r, ϕk) = (r, ϕ
(1)k , ϕ
(1)k , . . . , ϕ
(n−1)k ), k = 1, N —
сферичнi координати точок xk. Розглянемо iнтерполяцiю випадкового поляξ(x) в центрi сфери Sn за спостереженнями в точках xk, k = 1, N на сферi.
Розглядається випадок рiвномiрного розподiлу точок множиниX = x1, . . . , xN на сферi Sn . Вивчається асимптотична поведiнка похибкиiнтерполяцiї в цьому випадку. У зв’язку з цим отримано оптимальнийоб’єм N , починаючи з якого множина спостережень X = x1, . . . , xN єефективною.
Лiтература
[1] Карташов М.В. Скiнченновимiрна iнтерполяцiя випадкового поля наплощинi // Теорiя ймовiрн. i матем. статист. – 1994. – 51. – С.53-61.
[2] Semenovska N. Intepolation of a homogeneous and isotropic random field //Abstracts of international conference Modern Problems and New Trends inProbability Theory. II. – Chernivtsi, 2005. – P.96
[3] Ядренко М.И. Спектральная теория случайных полей. – Киев: Выщашкола, 1980. – 208с.
Київський нацiональний унiверситет iм.Тараса Шевченка,механiко-математичний факультет,кафедра теорiї ймовiрностей та математичної статистики,Вул. Володимирська, 64, 01033 Київ, Українаe-mail: [email protected].
77
Точнiсть та надiйнiсть моделi розв’язку задачi проколивання однорiдної струни.
Сливка Ганна Iванiвна, Тегза Антонiна Мигалiвна, Україна
В роботi запропоновано новий метод побудови моделi (яку можна реалiзуватина комп’ютерi) розв’язку задачi про коливання однорiдної струни зпочатковими умовами.
Розглянемо задачу про вiльнi коливання однорiдної струни, якщопочаткове вiдхилення точок струни i їх початкова швидкiсть є незалежнiстрого субгауссовi випадковi процеси, а кiнцi нерухомо закрiпленi:
uxx − utt = 0, x ∈ [0, l] t ∈ [0, T ], (1)
ut=0 = ξ(x),∂u
∂t
˛t=0
= η(x), (2)
u|x=0 = 0, u|x=l = 0. (3)
Розв’язок задачi (1)-(3) записується у виглядi
u(x, t) =
∞Xk=1
„Ak cos
„kπ
lt
«+Bk sin
„kπ
lt
««sin
„kπ
lx
«, (4)
де
Ak =2
l
lZ0
ξ(x) sin
„kπ
lx
«dx, Bk =
2
kπ
lZ0
η(x) sin
„kπ
lx
«dx.
Нехай модель процесу (4) має вигляд
uN (x, t) =
NXk=1
„Ak cos
„kπ
lt
«+ Bk sin
„kπ
lt
««sin
„kπ
lx
«, (5)
де
Ak =2
l
lZ0
ξ(x) sin
„kπ
lx
«dx, Bk =
2
kπ
lZ0
η(x) sin
„kπ
lx
«dx.
Отримано умови при яких випадковий процес uN (x, t) є моделлю, щонаближає випадковий процес u(x, t) з надiйнiстю 1 − γ та точнiстю δ врiвномiрнiй метрицi.
Розглянуто частинний випадок, коли η(x) = 0 , l = π , T = π , ξ(x) —
гауссiв випадковий процес такий, що ξ(x) =∞Pj=1
ξj sin(jx) , де ξj — незалежнi
нормально розподiленi випадковi величини такi, що Eξj = 0 , Eξ2j = bj , де b— деяке число таке, що 0 < b < 1.
78
Нехай ξ(x) = ξM (x) =MPj=1
ξj sin(jx). Використовуючи отриманi умови при
заданому Λ M вибрано так, щоб виконувалась нерiвнiсть
bM+1
1− b< Λ, тобто M ≥ ln(Λ(1− b))
ln(b).
В цьому випадку Ak = 2π
πR0
ξ(x) sin(kx) dx = ξk тобто, EA2k = bk. Отже,
AN = 2(√b)N+1(N(1−
√b) + 1)
(1−√b)2
+ 4ΛN
»(N + 1)2
4+ 1
– 12
,
ε0(N) ≤ (√b)N+1
1− b+ 2Λ
`N + (1 + ln(N))2
´ 12 = ε0(N).
Таким чином, шуканою моделлю буде модель, де N та Λ пiдiбранi так, щовиконуються нерiвностi
ANε0(N) <
„δ
2√π
« 12
та
0@δ − 3 · 223 δ
13 π
13A
13N (ε0(N))
13
ε0(N)
1A ≥ 2 ln
„1
γ
«.
Розв’язавши данi нерiвностi, ми отримали, що при Λ = 0, 0005 i N = 34модель uN (X, t) наближає випадковий процес u(X, t) з надiйнiстю 0, 99 таточнiстю 0, 01 в рiвномiрнiй метрицi областi D.
Лiтература
[1] Сливка Г. I., Тегза А.М. Моделювання розв’язку задачi коливанняоднорiдної струни з випадковими початковими умовами// Науковийвiсник Ужгродського ун-ту. Сер. матем. i iнформ. - Вип. 10-11.- 2005р,-С. 131-136.
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Близькiсть функцiй розподiлу двох сум випадковихвеличин
Слюсарчук Петро Володимирович, Боярищева Тетяна Валерiївна, Україна
Задача про близькiсть розподiлiв двох сум випадкових величин поставленав роботi [1], такi ж задачi розглядалися i в [2]. Робота [3] мiстить оцiнкунаближення розподiлiв сум представником iз певного класу розподiлiв.У данiй роботi продовжуються дослiдження розпочатi в [4], [5]. Нехайξ1, . . . ξn, . . . i η1, . . . ηn, . . . — двi послiдовностi випадковихвеличин вiдповiдно з функцiями розподiлу Fi(x) i Gi(x) ; характеристичнимифункцiями fi(t) i gi(t) , Φn(x) Qn(x) — вiдповiдно функцiї розподiлувипадкових величин
Pni=1 ξi Σni=1ηi Hi = Fi(x)−Gi(x) . Розглянемо наступнi
умови: iснує число α ∈ (0, 2] i стала λ > 0 такi, що
|gk(t)| ≤ e−λ|t|α
; (1)
µik =
Z +∞
−∞xkdHi(x) = 0 (i = 1, 2, . . . , k = 0, 1, 2, . . . , m), (2)
де
m =
1, α ≤ 1,2, x > 1.
Введемо псевдомоменти
χ(1)i (y) =
Z|x|≤y
|x|α|Hi(x)|dx, χ(2)i (y) =
Z|x|>y
|x|m−1|Hi(x)|dx,
χ(1)(y) = max1≤i≤n
χ(1)i (y), χ(2)(y) = max
1≤i≤nχ
(2)i (y);
χ(1)i0 (y) =
Z|x|≤y
max(1, |x|α)|Hi(x)|dx, χ(2)i0 (y) =
Z|x|>y
max(1, |x|m−1)|Hi(x)|dx,
χ(1)0 (y) = max
1≤i≤nχ
(1)i0 (y), χ
(2)0 (y) = max
1≤i≤nχ
(2)i0 (y);
ν(1)i0 (y) =
Z|x|≤y
max(1, |x|α+1)dHi(x), ν(1)0 (y) = max
1≤i≤nν
(1)i0 (y);
ν(2)i0 (y) =
Z|x|≤y
max(1, |x|m)dHi(x), ν(2)0 (y) = max
1≤i≤nν
(2)i0 (y).
Нехай ρn = supx|Φn(x)−Qn(x)|.
Теорема 1 Нехай виконуються умови i . Для будь-якого n ∈ n
ρn ≤ C(1) infy>0
χ(1)(y)
n1α
+ χ(2)(y) +1
αmax
“χ(y), (χ(y))
nn(α+1)+1
”ff,
80
ρn ≤ C(2) infy>0
(χ
(1)0 (y)
n1α
+ χ(2)0 (y) +
1
αmax
“χ0(y), (χ0(y))
nn+1
”),
ρn ≤ C(3) infy>0
(ν
(1)0 (y)
n1α
+ ν(2)0 (y)
),
деχ(y) max
“χ(1)(y); χ(2)(y)
”,
χ0(y) max“χ
(1)0 (y); χ
(2)0 (y)
”,
Cj , j = 1, 2, 3 – сталi, що залежать тiльки вiд α i λ .
Лiтература
[1] Золотарев В.М. О близости распределений двух сумм независимыхслучайных величин // Теория вероятностей и ее применение. – 1965. –Т. 10. Вып. 3. – с.519–526.
[2] Золотарев В.М. Современная теория суммирования независимыхслучайных величин. – М.: Наука. – 1986. – 416с.
[3] Нагаев С.В. Оценка погрешности приближения устойчивыми законами// Теорiя ймовiрностей та математична статистика. 1997. Вип. 56. – С.145 –160.
[4] Борищева Т.В., Слюсарчук П.В. Оцiнка близькостi розподiлiв двохсум для рiзно розподiлених випадкових величин // Науковий вiсникУжгородського унiверситету. Серiя математика та iнформатика. – 2001.– Вип. 6. – С. 4 – 8.
[5] Борищева Т.В., Слюсарчук П.В. Оцiнка близькостi розподiлiв сумвипадкових величин // Вiсник Київського унiверситету. Сер. фiз.-мат.науки – Київ, 2002. – Вип. 5. – С. 27– 32.
Ужгородський нацiональний унiверситет,88000, Ужгород, вул. Пiдгiрна, 46e-mail: kafmatan @ univ. uzhgorod. ua
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Зворотнi стохастичнi диференцiальнiрiвняння з Пуасонiвською компонентою та їх
застосування в фiнансовiй математицiОлена Соловейко, Україна
В фiнансовiй математицi широко застосовуються зворотнi стохастичнiдиференцiальнi рiвняння. В статтi [1] отримано багато результатiв у випадку,коли цiни акцiй є неперервними, тобто коли на ринку немає пуассонiвськоїкомпоненти. Виявляється, що результат про збiжнiсть можна отримати i увипадку, якщо рiвняння мiстить процес Пуассона.
Нехай задано повний iмовiрнiсний простiр (Ω, F, P ) , Rd - значнийброунiвський рух W , N - скалярний процес Пуассона.
Iснування i єднiсть розв’язку зворотного стохастичного диференцiальногорiвняння вигляду(
−dxt = f(ω, t, xt, qt, pt)dt− qtdWt − ptdNt,
xT = X, t ∈ [0, T ](1)
чи, еквiвалентно,
xt = X +
TZt
f(ω, s, xs, qs, ps)ds−TZt
qsdWs −TZt
psdNs, t ∈ [0, T ], (2)
встановлено в [2]. Тут Nt = Nt −tR0
λsds - скомпенсований мартингал
одновимiрного процесу Пуассона, iнтенсивнiсть процесу Пуассона λtвважається неперервною додатною функцiєю; кiнцеве значення — це Ft -вимiрна випадкова величина X : Ω → Rn ; породжуюча функцiя f євимiрною; вектор p : Ω × [0, T ] → Rn та матриця q : Ω × [0, T ] → Rn×d єпередбачуваними процесами; вектор x : Ω× [0, T ] → Rn - узгоджений процес.
Знайдено деякi апрiорнi оцiнки для розв’язку СДР типу (1) чи (2) та заїх допомогою доведено непевнiсть цих розв’язкiв за параметром.
Застосувавши отриманий результат про збiжнiсть розв’язку зворотногоСДР з пуассонiвською компонентою на фiнансовому ринку з однимбезризиковим та n ризиковими активами (акцiями), отримано збiжнiсть цiниплатiжного зобов’язання i збiжнiсть портфеля iнвестицiй за параметром.
Лiтература:1. El Karoui N., Peng S. and Quenez M.C. Backward stochastic differential
equations in finance // Mathematical Finance. - 1997. - 7:1. - p. 1-71.2. Rong S. Option Pricing in Mathematical Financial Market with Jumps and
Related Problems // Vietnam Journal of Mathematics. - 2002. - 30:2. - p.103-112.
Київський нацiональний унiверситет iменi Тараса Шевченка,Володимирська 64, Київ, 01033 Українаe-mail: [email protected]
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Двохстороннi, обмеженi розв’язкистохастичних системОлександр Станжицький, Україна
Розглядається система стохастичних рiвнянь Iто з малим параметром ε > 0
dx = εa(t, x)dt+√εb(t, x)dW (t), (1)
t ∈ R , x ∈ Rn , W (t) —вiнерiвський процес на R .Для таких систем обгрунтовано принцип усереднення i на його
основi одержанi умови iснування в рiвняння (1) обмежених в середньомуквадратичному двохстороннiх розв’язкiв.
Теорема 1 Нехай виконанi умови: 1) вектори a(t, x) , b(t, x) неперервнi поt ∈ R , x ∈ Rn i задовольняють умову Лiпшiца по x ; 2) |a(t, 0)|+|b(t, 0)| ≤ K ,K > 0 ; 3) iснують вектори a0(x) i b0(x) , що для деякої невiд’ємної функцiїα(T ) , α(T ) → 0 T →∞ виконанi нерiвностi:
1
T
˛˛TZ
0
[a(t, x)− a0(x)]dt
˛˛ ≤ α(T )(1+|x|), 1
T
˛˛TZ
0
[b(t, x)− b0(x)]dt
˛˛2
≤ α(T )(1+|x|2).
Тодi для довiльних η > 0 i T > 0 iснує ε0 > 0 , що при ε < ε0
E|x(t)− y(t)|2 < η, при t ∈»0,T
ε
–,
де x(t) (x(0) = x0) i y(t) (y(0) = x0) i вiдповiднi розв’язки рiвнянь (1) iусередненого рiвняння
dy = εa0(y)dt+√εb0(y)dW (t). (2)
Отримано результат про iснування двохстороннього, обмеженого всередньому квадратичному розв’язку системи (1).
Теорема 2 Нехай виконанi умови теореми 1 з замiню умови 3) умовою:рiвномiрно по t ∈ R справедливi нерiвностi
1
T
˛˛t+TZt
[a(s, x)− a0(x)]ds
˛˛ ≤ α(T )(1+|x|), 1
T
˛˛t+TZt
[b(s, x)− b0(x)]ds
˛˛2
≤ α(T )(1+|x|2).
Усереднена система (2) має рiвномiрно асимптотично стiйкий всередньому квадратичному розв’язок y = y0 . Нехай також виконанi умови:iснують M > 0 , A > 0 i γ > 0 , що при s ≤ t ≤ −M виконанi нерiвностi:
tRs
|a(u, x)− a0(x)|2du ≤ Ae−γt2(t− s)(1 + |x|2),
tRs
|b(u, x)− b0(x)|2du ≤ Ae−γt2(t− s)(1 + |x|2).
Тодi для ∀η > 0 , ∃ε0 > 0 , що при ε < ε0 система (1) має розв’язок x(t) ,визначений на R , i справедлива нерiвнiсть E|x(t)− y0|2 < η, t ∈ R.
e-mail: [email protected]
83
Мультифрактальний аналiз сингулярнонеперервних ймовiрнiсних мiр та його
застосування
Г.М.Торбiн∗ , Україна
В доповiдi аналiзуються взаємозв’язки рiзних пiдходiв до означенняхаусдорфової розмiрностi сингулярних ймовiрнiсних мiр на основiфрактального аналiзу суттєвих носiїв цих мiр. Введено в розглядхарактеристичнi мультифрактальнi мiри першого та вищих порядкiв,на основi яких здiйснено мультифрактальний аналiз сингулярнихiмовiрнiсних мiр та доведено теореми про структурне представленнятаких мiр. Обговорюються застосування отриманих результатiв. Особливаувага придiлена їх застосуванню в метричнiй теорiї чисел (дослiдженняфрактальних властивостей множин частково та суттєво анормальних чисел),в спектральнiй теорiї самоспряжених операторiв (мультифрактальнийрозклад сингулярно неперервного спектра) та безпосередньо в фрактальнiйгеометрiї (побудова нових методiв обчислення розмiрностi).
Нехай µ — ймовiрнiсна мiра на R з топологiчним носiєм Sµ . Надалiпiд словом ”розмiрнiсть” розумiтимемо розмiрнiсть Хаусдорфа-Безиковича.Нехай Aµ = E : E ∈ B, µ(E) = 1 — множина всеможливихборелiвських носiїв мiри µ . Хаусдорфовою розмiрнiстю мiри µ називаютьчисло dimH µ = inf
E∈Aµ
α0(E).
Означення. Локальною розмiрнiстю мiри µ в точцi x0 називаєтьсячисло
dimH(µ, x0) = limε→0
»inf
E∈Aµ
α0(E ∩ (x0 − ε, x0 + ε))–.
Мiра µ називається мiрою зовнiшньо точної розмiрностi α0 , якщо длядовiльної точки x0 ∈ Sµ виконується умова:
dimH(µ, x0) = α0 = dimH µ.Означення. Мiра µ називається сингулярною вiдносно хаусдорфової мiри
Hα порядку α (α -сингулярною), якщо iснує носiй A мiри µ нульової Hα -мiри. Мiра µ називається α -неперервною, якщо ∀A ∈ B з умови Hα(A) = 0випливає умова µ(A) = 0 .
Мiра µ називається мiрою внутрiшньо точної розмiрностi α , якщо длядовiльного ε ∈ (0, α) мiра µ є одночасно (α − ε) -неперервною i (α + ε) -сингулярною.
Всяка мiра внутрiшньо точної розмiрностi α має також зовнiшньо точнурозмiрнiсть.
Теорема 1 Якщо для мiри µ iснує множина A така, що µ(A) = 1 i ∀x ∈ A :α0(µ, x) := lim
δ→0
lnµ(x−δ,x+δ)ln δ
= α , то мiра µ є мiрою внутрiшньо точної
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розмiрностi α , множина A є мiнiмальним розмiрнiсним носiєм мiри µ(тобто dimH(µ) = α0(A)) i α0(A) = α .
Означення Мiра µ∗ , означена рiвнiстю
µ∗(E) := µ
x : lim
δ→0
lnµ(x− δ, x+ δ)
ln δ∈ E,∀E ∈ B
ff, (1)
називається характеристичною мультифрактальною мiрою (х.м.м.) першогопорядку для сингулярної мiри µ .
Очевидно, що мiра µ∗ має вироджений розподiл з атомом одиничної вагив точцi α0 тодi i тiльки тодi, коли мiра µ має внутрiшньо точну розмiрнiстьα0 .
Означення. Сингулярну мiру µ назвемо мiрою з дискретним(неперервним, абсолютно неперервним, сингулярно неперервним)мультифрактальним розподiлом, якщо вiдповiдна х.м.м. µ∗ є чистодискретною з бiльш нiж одним атомом (неперервною, абсолютнонеперервною, сингулярно неперервною) мiрою .
Теорема 2 Довiльна сингулярна iмовiрнiсна мiра µ єдиним чином можебути представлена у виглядi
µ = αac · µac + αsc · µsc +Xk
αk · µk,
де αac ≥ 0 , αsc ≥ 0 , αk ≥ 0 , αac + αsc +Pk
αk = 1 ; µk
— сингулярнi мiри внутрiшньо точної розмiрностi αk , µac (µsc ) —сингулярна iмовiрнiсна мiра з абсолютно неперервним (сингулярним)мультифрактальним розподiлом.
Теорема 3 Для довiльної ймовiрнiсної мiри ν на [0, 1] iснує сингулярнаiмовiрнiсна мiра µ , для якої характеристична мультифрактальна мiрапершого порядку µ∗ спiвпадає з ν.
Лiтература
[1] Albeverio S., Pratsiovytyi M., Torbin G., Topological and fractal propertiesof subsets of real numbers which are not normal. Bull.Sci.Math., 129 (2005),no. 8, 615–630.
[2] S.Albeverio, M.Pratsiovytyi, G.Torbin, Singular probability distributionsand fractal properties of sets of real numbers defined by the asymptoticfrequencies of their s-adic digits, Ukr.Math.J., 57 (2005), no.9, 1361-1370.
[3] Торбiн Г.М., Мультифрактальний аналiз сингулярно неперервнихймовiрнiсних мiр, Укр. Матем. Журн., 57 (2005), no.5, 837-857.
НПУ iменi М.П.Драгоманова, Україна, Київ
∗Partially supported by Alexander von Humboldt Foundation.
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Планування та аналiз вибiркових обстеженьна базi script-програм SPSS
Товмаченко Н.М., Сiчкар I.М., УкраїнаАктуальнiсть проблеми практичного використання планiв вибiркових
обстежень для отримання оцiнок середнього та загального значенняхарактеристик генеральної сукупностi вимагає розробки програмногозабезпечення побудови вiдповiдних оптимальних планiв та їх статистичногоаналiзу [1]. Метою даної роботи є розробка скрiпт-програм в середовищiпакета статистичних програм SPSS, якi будують оптимальнi плани зарiзними методами та виконують статистичний аналiз побудованих планiв[2]. Реалiзовано наступнi методи: 1) метод пропорцiйного стратифiкованогоопитування в умовах однорiдностi дисперсiй з одинаковими затратамивибiркових обстежень по стратумам; 2) метод стратифiкованого опитуванняз розрахунком об’єму стратумiв за критерiєм Неймана з неоднорiднимидисперсiями i одинаковими затратами вибiркових обстежень по стратумам;3) метод стратифiкованого опитування з розрахунком оптимального об’ємустратумiв з неоднорiдними дисперсiями та рiзними затратами вибiрковихобстежень по стратумам [3], [4].
При пропорцiйному стратифiкованому опитуваннi генеральнасукупнiсть, яка складається з N одиниць, спочатку дiлиться на Lстратумiв, якi складаються вiдповiдно з N1, N2, ..., NL одиниць, причомуN1 + N2 + ... + NL = N. Коли стратуми визначенi, вибiрка витягається зкожного стратума, де вiдбiр у рiзних стратумах виконується незалежноза схемою простого випадкового опитування. Об’єми вибiрок всерединiстратумiв позначаються вiдповiдно через n1, n2, ..., nL . Тодi для стратумаh : nh - кiлькiсть одиниць у вибiрцi; yhi− значення, отримане для i-оїодиницi, Wh = Nh
N-вага cтратума; fh = nh
Nh- частка вiдбору у стратумi.
При стратифiкованому опитуваннi оцiнки мають наступний вигляд:XStr =
PLh=1Whxh - оцiнка стратифiкованого середнього, xh -оцiнка
середнього в стратумi h, bXstr =PLh=1
bXh =PLh=1Nhxh – оцiнка загального.
Тодi точнiсть оцiнювання:
εxβ = z 1−β2
qDXstr = z 1−β
2
qPLh=1W
2h
1−fhnh
s2h,
де s2h = 1nh−1
Pi∈Plh
(xi − xh)2, P lh− сукупнiсть вибiркових одиниць запланом стратума, z 1−β
2− квантiль нормального розподiлу. Для оптимiзацiї
часу обчислень за скрiптом, розрахунок дисперсiї проводили за формулою:
s2k =Pni=1
(xki)2
nk−“Pn
i=1xkink
”2
. Довiрчi iнтервали мають вигляд:
Ixβ =˘x− εxβ , x+ εxβ
¯для середнього, Ixβ = NIxβ−для загального.
В першому алгоритмi об’єм вибiркових обстежень стратума пропорцiйнийоб’єму одиниць спостережень генеральної сукупностi в стратумах таприпускається, що затрати однаковi для вибiркових обстежень за всiмастратумами. В другому алгоритмi об’єм вибiркових обстежень стратумаформується за алгоритмом оптимального розподiлу Неймана:
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nh = nNhSyUh(PLh=1NhSyUh)−1.
Отже розмiр вибiрки у стратумi h повинен бути пропорцiйним NhSyUh . Вдеяких випадках формула дає nh > Nh . В такому випадку розмiр вибiрки устратумi h дорiвнює розмiру генеральної сукупностi у стратумi h: nh = Nh,а вищевказану формулу використаємо на залишенi стратуми з розмiромвибiрки nh−Nh . В третьому алгоритмi об’єм вибiркових обстежень стратумаформується за формулою:
nh0 = nWh0Sh0p
Ch0
(
LXh=1
WhSh√Ch
)−1, де Ch − витрати.
Статистичний аналiз побудованих вибiркових планiв проводиться закомплексом розроблених авторами синтаксис-програм та скрипт-програмв середовищi SPSS, якi дозволяють обчислити оцiнки середнього тазагального та їх довiрчi iнтервали для стратифiкованої випадкової вибiрки.Точнiсть оцiнювання характеристик генеральної сукупностi за вибiрковимиобстеженнями залежить вiд наявностi апрiорної iнформацiї. Якщо апрiорiвiдсутня iнформацiя щодо дисперсiй в стратумах та затрат на проведеннявимiрювань, а вiдомо лише розмiр стратума генеральної сукупностi, тодiслiд використовувати метод пропорцiйного стратифiкованого опитування вприпущеннях однорiдностi дисперсiй та рiвних затрат вибiркових обстеженьдля всiх страт. Цей метод оцiнювання має найгiршу вiдносну точнiсть оцiнкизагального, оскiльки обмаль апрiорної iнформацiї. За умови неоднорiдностiдисперсiй доречно застосовувати метод стратифiкованого опитування зрозрахунком об’єму стратумiв за критерiєм Неймана з одинаковимизатратами вибiркових обстежень по стратумам. Ця апрiорна iнформацiявiдносно дисперсiй характеристики за стратумами дозволяє пiдвищититочнiсть оцiнювання загального. Нарештi, якщо вiдомi апрiорнi оцiнкидисперсiй та затрат вимiрювання по кожному стратуму, то оптимальнимиє плани побудованi за методом стратифiкованого опитування з розрахункомоптимального об’єму стратума в умовах неоднорiдностi дисперсiй та рiзнихзатрат вибiркових обстежень по стратумам.
Лiтература[1] Tovmachenko N.M.,Sichkar I.M.,Vakulovych O.A.,Zhujkova E.M Design and analy-
sis of sample survey for small-scale enterprises.: International Workshop Predictionand decision making under uncertainties (PDMU-2004) Ternopil, 2004
[2] Ахим Бююль, Петер Цефер. SPSS: искусство обработки информации. Анализстатистических данных и востановление скрытых и восстановление скрытыхзакономерностей. Москва, Санкт-Петербург, Киев : торгово-издательский домDiaSoft, 2002 г.
[3] Cochran, William G. Sampling techniques, 3 ed. N.Y.: John Wiley & Sons, 1977[4] Lohr S.L. Sampling: design and analysis. - New York: Duxbury press., 1999.
Kyiv National Taras Shevchenko University, Faculty of Cybernetics,Department of Applied Statistics, Glushkov Avenue 2, building 6,Kyiv, 03680, Ukraine; e-mail: [email protected]
87
Оцiнки для розподiлу супремумастацiонарного квадратично гауссового
випадкового процесу
Федорянич Тетяна Василiвна, Україна
Розглядається задача оцiнювання розподiлу супремума стацiонарнихквадратично гауссових випадкових процесiв визначених на R+ .
Нехай X = X(t), t ≥ 0 -стацiонарний в широкому розумiннi квадратичногауссовий випадковий процес та для всiх t, s ≥ 0,
(E(X(t)−X(s))2)12 = eσ(|t− s|),
де eσ(|t− s|) < σ(|t− s|),σ = σ(u), u > 0 строго монотонно зростаюча неперервна функцiя,σ(u) −→ 0 при u −→ 0 та lim
u−→∞σ(u) = cσ <∞.
Розглянемо розбиття λ = t0, t1, . . . , tk, . . ., k = 1, 2, . . . множини R+ ,таке, що t0 = 0, tk−1 < tk, tk − tk−1 ≥ 1 та tk −→∞ при k −→∞ .
Нехай c(t) - деяка неперервна функцiя i 0 < c(t) < 1.
Позначимо Bk = [tk−1, tk], k = 1, 2, . . . , γk = supt∈Bk
c(t).
Введемо наступнi позначення:ε0k = inf
t∈Bk
sups∈Bk
m(t, s), δ0k = supt∈Bk
`E|X(t)|2
´1/2 , t0k = σ(ε0k) ,
z0k = max(δ0, t0),
Nk(u) – метрична масивнiсть Bk,
r(u) > 0, u ≥ 1 -монотонно зростаюча функцiя, r(u) −→∞ приu −→∞, така, що функцiя r(et) -опукла при t ≥ 0,
d =P∞k=1 γkz0k.
Основнi результати сформульованi в наступних теоремах.
Теорема 1 Нехай X = X(t), t ≥ 0 - стацiонарний сепарабельнийквадратично гауссiв випадковий процес та виконуються умови
1)P∞k=1 γk <∞,
2)P∞k=1 γk ln(tk − tk−1) <∞,
3) для деякого α > 0 та будь-якого ε > 0R ε0
“σ(−1)(u)
”−αdu <∞.
Тодi для довiльного x > 0 та 0 < p < 1 має мiсце нерiвнiсть
P
(supt∈R+
|c(t)X(t)| > x
)≤ 2 exp
−x(1− p)
d√
2
ff„1 +
√2x(1− p)
d
«1/2 eΦ1(p) ≤
88
≤ 2 exp
−x(1− p)
d√
2
ff„1 +
√2x(1− p)
d
«1/2 eΦ2(p),
де
eΦ1(p) = exp
(1
d
∞Xk=1
γkz0k ln
"„1
pt0k
Z pt0k
0
„1 +
tk − tk−1
2σ(−1)(v)
«αdv
«1/α#)
,
eΦ2(p) = exp
(1
d
∞Xk=1
γkz0k
»ln(tk − tk−1) +
1
αlnR∗(p)
–),
R∗(p) = supk
(„1
pt0k
Z pt0k
0
„1 +
1
2σ(−1)(v)
«αdv
« 1α
).
Теорема 2 Нехай X = X(t), t ≥ 0 - стацiонарний сепарабельнийквадратично гауссiв випадковий процес та виконуються умови
1)P∞k=1 γk <∞,
2)P∞k=1 γk ln(tk − tk−1) <∞,
3) для деякого α > 0 та будь-якого ε > 0R ε0
“σ(−1)(u)
”−αdu <∞.
Тодi для довiльного x > 0 та 0 < p < 1 має мiсце нерiвнiсть
P
(supt∈R+
|c(t)X(t)| > x
)≤ 2 exp
−x(1− p)
d√
2
ff„1 +
√2x(1− p)
d
«1/2 eΦ1(p) ≤
≤ 2 exp
−x(1− p)
d√
2
ff„1 +
√2x(1− p)
d
«1/2 eΦ2(p),
де
eΦ1(p) = exp
(1
d
∞Xk=1
γkz0k ln
"„1
pt0k
Z pt0k
0
„1 +
tk − tk−1
2σ(−1)(v)
«αdv
«1/α#)
,
eΦ2(p) = exp
(1
d
∞Xk=1
γkz0k
»ln(tk − tk−1) +
1
αlnR∗(p)
–),
R∗(p) = supk
(„1
pt0k
Z pt0k
0
„1 +
1
2σ(−1)(v)
«αdv
« 1α
).
Лiтература
1. Козаченко Ю.В., Федорянич Т.В. Оцiнки для розподiлу супремумаквадратично гауссових процесiв заданих на некомпактнихмножинах//Теорiя ймовiрностей та математична статистика. -2005. - 72. - С.64-77.
Ужгородський нацiональний унiверситет,88000, Ужгород, вул. Пiдгiрна, 46e-mail: [email protected]
89
Вычисление плотностей Радона-Никодимадля мер, порожденных решениями
некоторых нелинейных эволюционныхдифференциальных уравнений в
гильбертовом пространстве с заданнымикоэффициентами
T.А.Фомина, А.Д.Шаташвили, Украина
Пусть Ω,S, P – фиксированное вероятностное пространство, H –сепарабельное гильбертово пространство, со скалярным произведением(x, y), и нормой ‖x‖, где x, y ∈ H а L2 = L2[0, a], H обозначаетпространство функций, определенных на отрезке [0, a] со значениямииз H. L2 – гильбертово пространство со скалярным произведением(f, g)L =
R a0
(f(t), g(t)) dt и нормой ‖f‖2L =R a0‖f(t)‖2dt, f, g ∈ L2,
f(t), g(t) ∈ H.В пространстве H рассмотрим два эволюционных дифференциальных
уравнения: одно нелинейное, а другое, соответствующее к нему, линейное
dy(t)
dt−A(t)y(t) +A1(t)y(t) + αf(t, y(t)) = ξ(t),
0 ≤ t ≤ a, y(0) = ξ(0) = 0, (mod P)(1)
иdx(t)
dt−A(t)y(t) +A1(t)y(t) = ξ(t),
0 ≤ t ≤ a, x(0) = ξ(0) = 0, (mod P)(2)
где семейства A(t) и A1(t) являются, вообще говоря, неограниченнымиоператорами с одной и той же, не зависящей от t, областью определенияD(A) ⊆ H. Кроме того, операторное семейство A(t) является производящимоператором эволюционного семейства U(t, s) ограниченных операторов при0 ≤ t, s ≤ a, действующих в H, а семейство A1(t)U(t, s) при каждом0 ≤ t, s ≤ a ограничено, действует в H, и оба семейства U(t, s) и U1(t, s)интегрируемы со своим квадратом по норме H, ξ(t) – гауссовский случайныйпроцесс, Mξ(t) = 0, а его корреляционная операторная функция R2(t, s)в пространстве L2 как интегральное ядро, действующее при каждом t иs ∈ [0, a] в пространстве H порождает ядерный корреляционный операторR2 гауссовского случайного элемента ξ ∈ L2; нелинейная функция f(t, y(t))определена на [0, a] × H, принимает свои значения из H, интегрируема сосвоим квадратом по норме H для всех y(t) ∈ H, дифференцируема по y иее производная f ′y(t, y) для всех t ∈ [0, a] является оператором Гильберта-Шмидта, действующим в H. Обозначим через µy и µx меры, порожденныерешениями уравнений (1) и (2) соответственно, случайным процессом y(t) игауссовским процессом x(t), α – параметр.
90
В работе [1] нами были получены достаточные условия, при выполнениикоторых меры µy и µx эквивалентны (µx ∼ µy ) и в явном виде полученыформулы для вычисления плотностей Радона-Никодима dµy
dµx(z) и dµx
dµy(z).
В настоящей работе рассмотрены дифференциальные уравнения (1)и (2) с заданными коэффициентами A(t), A1(t), f(t, y), а также состационарным гауссовским процессом ξ(t), Mξ(t) = 0, и с заданнойкорреляционной функцией, например, R(t, s) = e−λ(t−s), удовлетворяющихусловиям теорем работы [1], и используя формулы для плотностейdµy
dµx(z) и dµx
dµy(z) как алгоритмы из [1], вычисляются их явные значения.
Полученные выражения плотностей уже могут быть использованы прирешении многих прикладных задач статистики случайных процессовдля рассматриваемых дифференциальных уравнений, в частности, дляполучения оптимальных оценок в задачах экстраполяции, фильтрации, атакже в задачах оптимального управления и, вообще, во многих задачахстатистического анализа. Кроме того, если управление (1) содержит малуюнелинейность (α – малый параметр), то тогда по известной методикеполученные оценки могут быть разложены по степеням малого параметрадо любой заданной точности, выделяя при этом в качестве главного членаразложения линейные оценки.
Литература
1. Т.А.Фомина, А.Д. Шаташвили, Плотности Радона-Никодима для мер,порожденных решениями нелинейных эволюционных дифференциальныхуравнений в гильбертовом пространстве H // Тезисы докладов намеждународной конференции International conference Modern problems andnew trends in probability theory, Chernivtsi, Ukraine, 2005, vol. 2, P. 117.
Донецкий государственный университет экономики и торговли,г.Донецк, ул. Щорса, 31
Институт прикладной математики и механики НАН Украины,отдел теории вероятностей и математической статистики,г. Донецк, ул. Р.Люксембург, 74
e-mail: [email protected]
91
Асимптотическое поведение одного классастохастических полугрупп
Чани А.С., Украина
Изучается асимптотическое поведение дискретных матричнозначныхстохастических полугрупп над полем комплексных чисел. В фазовомпространстве определенным образом выделяется семейство подалгебр.Основной результат состоит в следующем. Пусть стохастическая полугруппапостроена по распределению Бернулли: ее приращения принимают значенияА и В с вероятностью р и q соответственно. Если А и В принадлежатодной и той же подалгебре, то для стохастической полугруппы в явномвиде вычисляется ее спектральный радиус. Явный вид означает выражениеспектрального радиуса в терминах р и q и собственных чисел матриц Аи В. Полученный результат легко обобщается на случай полиномиальногораспределения.
Institute of Applied Mathematics and Mechanics NAS of Ukraine,R.Luxembrg street, 74e-mail: [email protected]
92
Об одном методе вычисления оптимальныхоценок в задачах экстраполяции для
решения эволюционных дифференциальныхуравнений в гильбертовом пространстве
А.А.Шаташвили, А.Д.Шаташвили, Украина
Пусть Ω,S, P – фиксированное вероятностное пространство, H –сепарабельное гильбертово пространство, (·, ·), ‖·‖ – скалярное произведениеи норма в H соответственно, S – σ -алгебра измеримых подмножествпространства H, x(t) и ξ(t) – случайные процессы, определенные на отрезке[0; a] и со значениями в H , а µx и µξ – их соответствующие меры,порожденные на σ -алгебре S.
Пусть случайный процесс x(t) наблюдается до момента T, T ∈ [0; a] итребуется вычислить его оптимальную оценку η = bx(T + h) (или прогноз)в точке T + h ∈ [0; a] в смысле минимума его среднеквадратическогоотклонения от его истинного значения в точке T+h, minη∈FT M(η−x(T+h)),где FT – σ -алгебра событий, определенная поведением случайного процессаx(t) при t ≤ T. Ясно, что η – FT -измеримая величина и известно,что (см. [1]) она вычисляется как условное математическое ожиданиеη = bx(T + h) = M(x(T + h)/FT ). В работе [2] предлагается метод, в которомвычисление условного математического ожидания сводится к вычислениюбезусловного математического ожидания. В этом случае доказываетсятеорема для получения алгоритма вычисления оптимального прогноза приусловии, что во-первых ξ(t) – гауссовский случайный процесс, а во-вторыхмеры µx и µξ – эквивалентны (µx ∼ µξ ) и известна их плотностьРадона-Никодима ρa(z) = dµx
dµξ(z) для всех t ∈ [0; a]. В указанной работе
результаты были получены для решения нелинейных дифференциальныхуравнений, возмущенных гауссовскими процессами и достаточно гладкимиусловиями на коэффициенты уравнений , а плотности Радона-Никодимавычислены в терминах стохастического интеграла Ито. В предлагаемойже работе в гильбертовом пространстве H рассматривается аналогичнаязадача, но в значительно более широких предположениях на коэффициентырассматриваемых уравнений с использованием плотности Радона-Никодима,вычисленной уже в терминах расширенного стохастического интеграла.В частности, в пространстве H рассматриваются два эволюционныхдифференциальных уравнения, одно нелинейное, а другое – линейное безнелинейной части с неограниченными линейными операторами
dy
dt−B(t)y(t) + αf(t, y(t)) = ξ(t), (1)
dx
dt−B(t)x(t)) = ξ(t), (2)
93
0 ≤ t ≤ a, y(0) = x(0) = ξ(0) = 0,
B(t) = A(t)−A1(t), (3)
где ξ(t) – гауссовский случайный процесс, α – параметр, A(t) и A1(t) –линейные неограниченные операторы с одной и той же не зависящей от tобластью определения D(A) ⊆ H, порождающие семейство ограниченныхэволюционных операторов U(t, s) и U1(t, s) , (0 ≤ t, s ≤ a), с достаточнымисвойствами, удовлетворяющие вместе с функцией f(t, y) условиям теоремиз работы [3], которые обеспечивают эквивалентность мер µy и µx,порожденных случайными процессами y(t) и x(t) – решениями уравнений (1)и (2), при которых в явном виде выписываются плотности Радона-Никодимапо формулам, полученных в той же работе [3] ρα(z) =
dµy
dµx(z), но уже в
терминах расширенного стохастического интеграла.Тогда, используя идеи, предложенные в работе [2], строится алгоритм
для вычисления оптимального прогноза экстраполяции функции y(t) вточке t=(T+h), при условии, что наблюдается ее поведение за время t ≤ T.Полученная формула обобщает результаты, полученные ранее в работе [2] иприменима для решений класса нелинейных дифференциальных уравненийпри более широких предположениях на его коэффициенты, чем в указаннойработе [2].
Литература
1. И.И.Гихман, А.В.Скороход Введение в теорию случайных процессов.М.: Наука , 1977, 568 с.
2. А.Д. Шаташвили Оптимальная экстраполяция и фильтрация дляодного класса случайных процессов, II. // Теория вероятностей иматематическая статистика. – 1970. вып. 3 с. 211-231.
3. Т.А.Фомина, А.Д. Шаташвили, Плотности Радона-Никодима для мер,порожденных решениями нелинейных эволюционных дифференциальныхуравнений в гильбертовом пространстве H // Тезисы докладов намеждународной конференции International conference Modern problems andnew trends in probability theory, Chernivtsi, Ukraine, 2005, vol. 2, P. 117.
Институт прикладной математики и механики НАН Украины,отдел теории вероятностей и математической статистики,г. Донецк, ул. Р.Люксембург, 74e-mail: [email protected]
94
О некоторых упражнениях по теориивероятностей, предложенных М.И. Ядренко
Тенгиз Шервашидзе, Грузия
В скромный знак памяти о М.И. Ядренко
В третьем издании учебника Б.В. Гнеденко [1] появились три упражненияпредложенные М.И. Ядренко: 29–30 к главе 4: Случайные величиныи функции распределения, и 22 к главе 5: Числовые характеристикислучайных величин. В трех следующих изданиях содержатся и другие задачи,предложенные М.И. Ядренко: 31–32 к главе 4 и 23 к главе 5. Этизадачи, кроме 32 гл. 4 (которая по-видимому исходит от Чубера [2]; мы неставим целью указать первоисточники других задач), вошли в сборник задач[3] под номерами II.3.104, II.3.102, II.3.84, III.2.26 и II.3.59, соответственно.
Все задачи подобраны с тонким вкусом.По опыту автора при преподавании весьма полезно использовать
следующие три из перечисленных задач:1 ([4], гл. 4, 30). Случайные величины ξ и η независимы. Их плотности
распределения соответственно равны
pξ(x) = (πp
1− x2)−1, |x| < 1 (= 0, |x|≥1), pη(x) = xe−x2/2, x < 0 (= 0, x≤0).
Доказать, что величина ξη нормально распределена.2 ([1, кроме 3-его изд.], гл. 4, 31). Пусть ξ и ζ незавсимы и имеют
плотности распределения pξ(x) = pζ(x) = λe−λx , x ≥ 0 (= 0 , x < 0) .Доказать, что η = ξ/(ξ + ζ) распределена равномерно на отрезке (0, 1) .
3 ([1], гл. 5, 22). Случайные величины ξ1, ξ2, . . . равномернораспределены в (0, 1) . Пусть ν – случайная величина, равная тому k , прикотором впервые сумма sk = ξ1 + · · · + ξk превосходит 1. Доказать, чтоMν = e .
Первую задачу хорошо рассматривать вместе с задачей 23 из [1], гл.4 (задача II.3.103 из [3]), предлагающей доказать, что если случайныевеличины независимы и нормальны (0, σ2) , то величины ξ2 +η2 и ξ/η такженезависимы. Эти две задачи дают информацию о возможности представитьнормальное распределение на плоскости с центром в начале координати круговой симметрией, путем перехода к полярным координатам, какраспределение пары независимых случайных величин с распределением χ2
и равномерным распределением в (0, 2π ), соответственно.Задача 2 дает простую связь между показательным и равномерным
распределениями (аналогичная связь существует, например, между паройнезависимых случайных величин с распределениями χ2
m и χ2n и β -
распределением) и может быть переформулирована как задача теориимассового обслуживания.
95
Задачу 3, наиболее красивую по мнению автора из трех приведенныхвыше задач, можно использовать в качестве основы для иллюстрации законабольших чисел и метода Монте-Карло для статистического приближениячисла e наподобие классической задачи Бюффона в связи с приближениемчисла π .
Книги [3] и [4] свидетельствуют о большом вкладе М.И. Ядренков разработку богатых по содержанию и методически сбалансированныхбазовых курсов теори вероятностей и математической статистики. В [4]имеется достаточно много примеров и упражнений, но [3] дает уникальнуюколлекцию самых разнообразных задач разной сложности. Хотелось быотметить, что среди них есть задачи, иллюстрирующие вероятностныеметоды решения задач математического анализа, в частности, теорииаппроксимации функций, важных для воспитания у студента единого взглядана математику. Ярким примером таких методов является классическоевероятностное доказательство теоремы Вейерштрасса устанавливающее спомощью неравенства Чебышева и закона больших чисел равномернуюсходимость полиномов Бернштейна к непрерывной на сегменте функции(задача III.4.25). Задача III.4.24 дает общие условия равномерной сходимостиматематического ожидания функции от сходящейся по вероятностипоследовательности случайных величин зависяшщих от действительногоаргумента. Из нее следуют решения задач III.4.25 – 27. Последняя дает ключк выводу формулы обращения для преобразования Лапласа, принадлежащуюУиддеру (задача III.4.28, см. также [4], стр.146-147).
Литература
[1] Б.В. Гнеденко, Курс теории вероятностей. 3-е изд. Гос. изд-во физ.-мат.лит., Москва, 1961; 4-е, 5-е, 6-е изд., Наука, Москва, 1965, 1969, 1988.Перевод на англ. яз. 6-го изд.: Theory of probability. Gordon and Breach,Newark, NJ, 1997.
[2] Em. Czuber, Geometrische Wahrscheinlichkeiten und Mittelwerte. Teubner,Leipzig, 1884.
[3] А.Я. Дороговцев, Д.С. Сильвестров, А.В. Скороход, М.И. Ядренко,Терия вероятностей. Сборник задач. Выща школа, Киев, 1980. Переводна англ. яз.: Probability theory: collection of problems. Translations of Math-ematical Monographs. 163. American Mathematical Society, Providence, RI,1997.
[4] И.И. Гихман, А.В. Скороход, М.И. Ядренко, Теория вероятностей иматематическая статистика. Выща школа, Киев, 1988.
Математический институт им. А Размадзе,ул. М. Алексидзе 1, Тбилиси 0193, Грузияe-mail: [email protected]
96
Мiнiмiкснi оцiнки функцiоналiв вiдоднорiдного випадкового поля
Наталiя Щестюк, Україна
Дослiджується задача оптимального лiнiйного оцiнювання функцiоналаAKξ =
P(k,j)∈K a(k, j)ξ(k, j) вiд однорiдного випадкового поля ξ(k, j) , що
має щiльнiсть f(λ, µ) , за даними спостережень поля ξ(k, j) + η(k, j)при (k, j) ∈ Z2\K , де η(k, j) однорiдне випадкове поле, що маєщiльнiсть g(λ, µ) . Область K ⊂ Z2 має вигляд нескiнченної смугиабо пiвплощини. Щiльностi розкладаються на множники, тобтоf(λ, µ) = f1 (λ) f2 (µ) ,g (λ, µ) = f1 (λ) g2 (µ) . Запропоновано формулидля обчислення спектральної характеристики та середньоквадратичноїпохибки оптимальної лiнiйної оцiнюнки функцiонала AKξ за умови, щощiльностi вiдомi точно. В умовах невизначеностi, коли точнi значенняf1(λ), f2(λ), g2(µ) невiдомi, а визначенi лише класи можливих щiльностей,застосовується мiнiмаксний пiдхiд, тобто такий, який мiнiмiзує максимальнувеличину похибки одночасно для всiх щiльностей з даного класу. Цей пiдхiдбув запропонований У. Гренадером i розвинутий в роботах М. Танiгушi,С. Кассама, Г. Пура, Ю. Франке. У цьому випадку знайденi формули длязнаходження найменш сприятливих спектральних щiльностей та мiнiмiксних(робастних) спектральних характеристик оцiнок функцiонала AKξ .
Мiнiмаксний метод застосовано до класу спектральних щiльностей.D = D2ε1(λ)×D2ε2(λ) , де
D2ε1(λ) =
8<:f (λ, µ)
˛˛ 1
2π
πZ−π
(f (λ, µ)− u1 (λ, µ))2 dµ ≤ ε1(λ) ,∀λ ∈ [−π, π]
9=; ,
D2ε2(λ) =
8<:g (λ, µ)
˛˛ 1
2π
πZ−π
(g (λ, µ)− u2 (λ, µ))2 dµ ≤ ε2(λ) ,∀λ ∈ [−π, π]
9=; .
1. Moklyachuk M.P., Shchestyuk N.Yu. Robust estimates of functionals of ho-mogeneous random fields, Theory of Stochastic Processes. – Vol. 10 (26),no.1–2, 2004, pp.196–209.
2. Kassam S.A., Poor V.H. Robust techniques for signal processing: A survey,Proceedings IEEE. - 1985.- Vol.73. No.3.- P. 433-481.
Схiдноукраїнський нацiональний унiверситет iм. В. Даля,факультет прикладної математики та iнформатики,кафедра математичного аналiзу,91034 Луганськ, квартал Молодiжний, 20-аe-mail: [email protected]
97
Necessary condition of optimality forstochastic systems with variable delay on
controlAgayeva C.A., Azerbaijan
It is considered optimal control problem with variable delay control in whichconstraints on the state variable are given with the help of arbirary convex closedsets. In the present paper is solved the problem with controlled diffusion.
Let (Ω, F, P ) be a probability space with filtration F t = σ(ws, t0 ≤ s ≤ t)L2F (t0, t1;Rn) .
Let wt be an n - dimensional Wiener process and F t = σ (ws, t0 ≤ s ≤ t) .L2F (t0, t1;Rn) is the space of all predictable processes x : [t0, t1]×Ω → Rn such
that Et1Rt0
|xt (ω)|2 dt < +∞ . Rm×n is the space of linear transformations from
Rm to Rn .Consider the following stochastic optimal control problem with variable delay
on control:
dxt = g(xt, ut, ut−h(t), t)dt+ f(xt, ut, ut−h(t), t)dwt, t ∈ (t0, t1] , (1)
xt0(ω) = x0, (2)
ut(ω) = S(t), t ∈ [t0 − h(t0)) , h(t) 0, (3)
ut ∈ Ud ≡˘ut(ω) ∈ L2
F (t0, t1;Rm)/u•(ω) ∈ U ⊂ Rm, a.c.¯
(4)
S(t) - is given non-random function, h(t) 0 is continuously differentiable non-random function such that dh(t)
dt< 1 . The problem is concluded in minimization
of cost functional:
J (u) = E
8<:p (xt1) +
t1Zt0
l (xt, ut, t) dt
9=; (5)
which determined on the decisions of the system (1) - (4), generated by all admis-sible controls at condition:
Eq(xt) ∈ G, (6)
G is convex closed set in .I. l (x, u, t) , g (x, u, ν, t) , f (x, u, ν, t) are twice continuously differentiable with
respect to x :g : Rn ×Rm ×Rm × [t0, t1] → Rn, l : Rn ×Rm × [t0, t1] → R ,f : Rn ×Rm ×Rm × [t0, t1] → Rn×n
II. l, g, f and all their derivatives are continuous in (x, u, t) and bounded:
(1 + |x|)−1 (|g (x, u, ν, t)|+ |gx (x, u, ν, t)|+ |l (x, u, t)|+ |lx (x, u, t)|++ |f (x, u, ν, t)|+ |fx (x, u, ν, t)|) ≤ N.
98
III. p : Rn → R is twice continuously differentiable :
|p(x)|+ |px(x)| ≤ N (1 + |x|) ; |pxx(x)| ≤ N.
IV. q : Rn → Rk is twice continuously differentiable :
|q(x)|+ |qx(x)| ≤ N (1 + |x|) ; |qxx(x)| ≤ N.
For optimality of a control, we present the following necessary condition.
Theorem. Assume that the assumptions I-IV are satisfied and`x0t , u
0t
´− op-
timal solution of problem (1) -(6). Then there are a nonzero (λ0, λ1) ∈ Rk+1 andrandom processes (ϕt, γt) ∈ L2
F (t0, t1;Rn) × L2F (t0, t1;Rnxn) ,
(Ψt, Zt) ∈ L2F (t0, t1;Rn)× L2
F (t0, t1;Rnxn) that are solutions of:8<:dϕt = −
ˆg∗x`x0t , u
0t , v
0t , t´ϕt + f∗x
`x0t , u
0t , v
0t , t´βt − λ0l
`x0t , u
0t , t´˜
++γtdwt, t ∈ [t0, t1) ,
ϕt1 = −λ0px (xt1)− λ1qx (xt1) ;8>>>><>>>>:dΨt = −[g∗x(x0
t , u0t , v
0t , t)Ψt + Ψtgx(x0
t , u0t , v
0t , t)+
+f∗x (x0t , u
0t , v
0t , t)Ψtfx(x0
t , u0t , v
0t , t)dt+
+f∗x (x0t , u
0t , v
0t , t)Zt + Ztfx(x0
t , u0t , v
0t , t)+
+Qxx(ϕt, x0t , u
0t , v
0t , t)]dt+ Ztdwt,t0 ≤ t < t1,
Ψt1 = −λ0pxx (xt1)− λ1qxx (xt1) ;
such that for all u ∈ U almost certainly holds:
Q`ϕθ, x
0θ, u, νθ, θ
´−Q
`ϕθ, x
0θ, u
0θ, νθ, θ
´+ˆQ`ϕz, x
0z, u
0z, u, z
´−Q
`ϕz, x
0z, u
0z, ν
0z , z´˜z=r(θ)
+
+ 12
ˆf∗`x0θ, u, ν
0θ , θ´− f∗
`x0θ, u
0θ, ν
0θ , θ´˜
×Ψt
ˆf`x0θ, u, ν
0z , θ´− f
`x0θ, u
0θ, ν
0θ , θ´˜
++ 1
2
ˆf∗`x0z, u
0z, u, z
´− f∗
`x0z, u
0z, ν
0z , z´˜
×Ψz
ˆf`x0z, u
0z, u, z
´− f
`x0z, u
0z, ν
0z , z´˜z=r(θ)
≤ 0
a.e. θ ∈ [t0, t1 − h) ,
Q(ϕθ, x0θ, u, vθ, θ)−Q(ϕθ, x
0θ, u
0θ, vθ, θ)+
+ 12∆uf
∗(x0θ, u
0θ, v
0θ , θ)Ψt∆uf(x0
θ, u0θ, v
0θ , θ) ≤ 0
a.e. θ ∈ [t1 − h, t1],here νt = ut−h(t),Q (ϕt, xt, ut, νt, t) = ϕ∗t g (xt, ut, νt, t) + γ∗t f (xt, ut, νt, t)− l (xt, ut, t) .
Institute of Cybernetics,Azerbaijan National Academy of Sciences,F.Agayev st.9, AZ1141 Bakue-mail: [email protected]
99
Necessary and sufficient condition ofoptimality for linear stochastic control systems
with variable structureAgayeva C.A., Abushov Q.U., Azerbaijan
It is considered linear optimal control problem for stochastic systems withvariable structure. In the present paper is solved the square linear control problem.We drive necessary and sufficient condition for this problem.
Let (Ω, F, P ) be a probability space with filtration˘F t, t ∈ [0, T ]
¯. Let
wt be an n -dimensional Wiener process and F t−σ (ws, 0 ≤ s ≤ t) . L2F (0, T ;Rn)
is the space of all predictable processes x : [0, T ] × Ω → Rn such thatER T0|xt(ω)|2 dt < +∞ . Rm×n is the space of linear transformations from Rm
to Rn .Let 0 = t0 < t1 < ... < ts−1 < ts = T and y = (y1, y2, ..., yn) , t ∈ [tl−1, tl) ,
l = 1, s . First we will consider the following stochastic optimal control problemwith variable structure:
dy =“Altyt +Bltut
”dt+ Cltytdwt (1)
y(tl) = xl l = 0, s− 1. (2)ult ∈ U l ⊂ Rm, a.s., l = 1, s (3)
The problem is concluded in minimization of cost functional:
I(0, y, u, T ) = 〈GyT , yT 〉+
Z T
0
[〈Mτyτ , yτ 〉+ 〈Nτuτ , uτ 〉] dτ (4)
bI = E [I(0, y, u, T )] (5)which determined on the decisions of the system (1)-(3), generated by all admis-sible controls U = U1 × U2 × ...× Us .
Let yn+1 = I(0, y, u, t) = 〈Gyt, yt〉+R t0
[〈Mτyτ , yτ 〉+ 〈Nτuτ , uτ 〉] dτ
dyln+1 =DGAltyt +Al∗t Gyt +Mtyt, yt
Edt+
DBl∗t G
∗yt +Bl∗t Gyt +Ntut, utEdt+
+DGGltyt +Gl∗t Gyt, yt
Edwt, l = 1, s (6)
yln+1(tl) = 〈Gytl ytl〉 l = 0, s− 1 (7)Our purpose: α(T ) = E
Psl=1 y
ln+1(tl) → min . (8)
Instead of problem (1)-(5) shall consider (n+ 1) -measured problem (1)-(3), (6)-(8). It is received necessary and sufficient condition of optimality for this problem,which is necessary and sufficient condition of optimality for source problem (1)-(5)too.Institute of Cybernetics, Azerbaijan National Academy of Sciences,F.Agayev st.9, Baku AZ1141 Baku, e-mail: [email protected]
100
Modelling of Galton-Watson branchingprocesses
Aliev S.A., Azerbaijan
Lately, important results allowing to estimate the behavior of different char-acter of the investigated processes are obtained on the investigation of branchingprocesses. However, mainly these results have asymptotic character and allow toconclude on the investigated characteristics at sufficiently long observation.
Wide use of computers in scientific investigations, modelling of branchingprocesses in computers allows to observe the behavior of branching processes onany interval. At that there arises a problem on the obtained data processing thatrequires to elaborate new universal methods of data analysis.
In the paper a method of the modelling and studying some characteristicsof Galton-Watson branching processes with further processing of the results ofmodelling is suggested.
At modelling the process ζn is considered and it is assumed that at initialtime these is one particle ζ0 = 1 and probabilistic low of population developmentis of the following form
pk = P ζ1 = k =1
2k, k = 1, 2...
By composing a program for modelling of the Galton-Watson branchingprocess in computer, the pseudorandom number is chosen from the interval (0, 1) .We’ll denote this random number by α . It is assumed that if pk+1 < α < pk ,then in the following generation the quantity of anew formed particles equals k ,if α > 1
2, then the number of descendants in the next generation equals 0 .
At such assumption the mean number of particles and dispersion of the numberof particles in the given generation the process are investigated.
It should be noted that the introduced algorithm of modelling of Galton-Watson branching processes is universal and allows to model more complicatedbranching processes
Institute of Mathematics and Mathematics of NAN of Azerbaijane-mail: soltan [email protected]
101
The estimation of the ruin probability in themodel with investments and variable premiums
Maryna Androshchuk, Ukraine
We consider a problem of the ruin probability estimation in the risk modelgiven by equation
(1) Rt(u,K) = u−PNk=1 Uk+
R t0p(Rs) ds+
R t0KsSsds+
R t0Rs−Ks
eδs deδs, t ≥ 0.
Here u > 0 is an initial capital of the insurance company. Nt, t ≥ 0 is aPoisson process with intensity β > 0 . A sequence of i.i.d. r.v. Uk is independentof Nt, t ≥ 0 , it models the size of claims incurred by the insurer. p(Rs) isthe process of incoming insurance premiums, it depends on the current reserve ofthe company. Ks is an amount of money invested in the stock, and the remainingreserve Rs −Ks is an amount invested in the bond.
Stock price is described by geometric Brownian motion(2) dS(t) = S(t)(adt+ bdWt),
where a > 0 , b > 0 are fixed constants and Wt is a standard Brownian motionindependent of the compound Poisson process.
We use results from [3] to prove an existence of a solution of equation (1).One of our results is the following. If the function p(x) is bounded; inequality
(3) r2b2
2K2t + r(δ − a)Kt − rp(Rt)− rRtδ + βh(r) ≤ 0 a.s., t ≥ 0,
is hold; some other (not very restrictive) conditions on the process Kt arehold, then the process e−rRt(u,K), t ≥ 0 is supermartingale w.r.t. the fil-tration generated by the risk process (1). In the case of equity in (3) the processe−rRt(u,K), t ≥ 0 is local martingale. In the proof we use techniques developedin [2].
For bounded p(x) , a > δ and constant investment process Kt = a−δrb2
, t ≥ 0,the ruin probability ψ(u,K) of an insurer can be bounded from above by
(4) ψ(u,K) ≤ e−ru,
where r is the only positive solution of the equation
(5) βh(r) = (δ−a)22b2
+ r infx≥0p(x) + xδ.This result generalizes the result obtained in [1], where an estimation of ruin
probability in the model with constant premium rate over time was found in thesame form.
[1] Gaier J., Grandits P., Schachermayer W. Asymptotic Ruin Probabilities andOptimal Investment, The Annals of Applied Probability , Vol 13, 2003, p.1054-1076. [2] Protter P.E. Stochastic Integration and Differential Equations,"Springer", Berlin, 2004, p. 410. [3] Kiffe T.A. Discontinuous Volterra IntegralEquation, J. Integral Equations , Vol 1, 1979, p. 193-200.
Kyiv National Taras Shevchenko University, Dept. of Mechanics and Mathematics,Volodimirska, 64, 01033, Kyiv, Ukraine. E-mail: andr_ [email protected]
102
About asymptotic behavior of the unit rootbilinear model with dependent noise sequence
Taras Androshchuk, Ukraine
The term of "bilinear model" we use for a random sequence Xk generatedby recurrent equation
Xk = aXk−1 + εk + bXk−1εk−1 (1)
where εk are i.i.d. random values with Varεk = σ2 <∞ . In the case of a = 1we call the model as a unit root bilinear model. In the recent time model (1) isvery popular for the description of different economics data. Indeed, equation (1)has only two parameters but it involves a big variety of very important classes ofprocesses. For example, if a = 1 , b = 0 we have a random work with independentincrements. For |a| < 1 , b = 0 we have autoregressive model, which is stationary.In general the process Xk is stationary under condition a2 + b2σ2 < 1 , but inthe case of b 6= 0 it is not Markov any more.
In model (1) we propose to use a noise sequence εk which is not independent.For that purpose we put εk = BHk+1−BHk , where BHt is a fractal Brownian motionwith Hurst parameter H > 1
2. As it was done in [1] we may assume that coefficient
b is not constant, and converges to zero in series of experiments which are indexedby n ≥ 1 . Namely, put b = b(n) = β
nH . Then, after scaling the process Xk as
X(n)t =
XknH
,k
n≤ t <
k + 1
n, 0 ≤ k ≤ [n · T ]− 1, (2)
we prove that the process X(n) ⇒ Y weakly in the Skorohod space D[0, T ] asn→∞ , where
Yt = σ
Z t
0
expnσβ`BHt −BHs
´odBHs (3)
is a solution of stochastic differential equation dYt = σ`1 + β · Yt
´dBHt , Y0 = 0 .
A similar result can be obtained in the case when for εk we use incrementsof random walk with kernel weights that approximates BHt (see [3]).
The work is made under the INTAS project 03-51-3714.
References
[1] Lifshits M.A. Invariance principle in a bilinear model with weak non-linearity, Zapiski Seminarov POMI, 2004, 320, 97–105 (in Russian).
[2] Sottinen T. Fractional Brownian motion, random walks and binary marketmodels, Finance and Stochastics 5, no. 3, 2001, pp. 343 – 355.
Kyiv National Taras Shevchenko University,Department of Mechanics and Mathematics,Volodimirska, 64, 01033, Kyiv, Ukrainee-mail: [email protected]
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Properties of the MLE in dynamical systemswith small fractional noise
Taras Androshchuk, Ukraine
We consider a problem of parameter estimation in the dynamical system givenby equation
Xt = x0 +
Z t
0
S(θ, u,Xu) du+ εBt, 0 ≤ t ≤ T, (1)
where Bt is a fractional Brownian motion with Hurst parameter H ∈ ( 12, 1) ,
θ is unknown parameter from a convex open set Θ ⊂ Rd , parameter ε > 0reflects intensity of the noise. The function S is taken from the class of linearfunctions S(θ, t, y) = f1(θ, t) + f2(θ, t) · y with sufficiently smooth coefficientsf1(θ, t) , f2(θ, t) .
Assume that the Fisher’s information matrix I(θ) which is built on the baseof the solution xt of deterministic equation (1) with ε = 0 is uniformly positivedefinite. We prove that a maximum likelyhood estimation θε (MLE) which is builton the trajectory observation X = X(ε)
(a) is consistent as ε→ 0
Pθ − limε→0
θε = θ; (2)
(b) is asymptotically normal as ε→ 0
Lnε−1I(θ)
12`θε − θ
´o→ N(0, I); (3)
(c) has convergent moments for all p > 0
limε→0
supθ∈K
˛Eθ˛I(θ)
12`θε − θ
´˛pε−p − |∆|p
˛= 0, (4)
where L∆ ' N(0, I) , I is a unique matrix d× d .The above result is obtained with a help of the theorem from the general
theory of asymptotic estimation due to Ibragimov and Has’minskii [1] and themethods developed by Kutoyants for the problem of parameter estimation in themodel with small standard Brownian noise [2].
The work is supported by the INTAS project 03-51-3714.
1. Ibragimov I.A., Has’minskii R.Z. Asymptotic theory of estimation,"Nauka", Moscow, 1979, p. 528.
2. Kutoyants Yu. A. Identification Of Dynamical Systems With Small Noise,Mathematics and Its Applications, Vol 300, Kluwer Academic Publisher,1994, p. 298.
Kyiv National Taras Shevchenko University,Department of Mechanics and Mathematics,Volodimirska, 64, 01033, Kyiv, Ukrainee-mail: [email protected]
104
Results on the fractal measure of some sets
Alina Barbulescu, Romania
The fractal dimensions are very important characteristics of fractal sets. Aproblem which arises in the fractal sets study is the determination of their dimen-sions. The Hausdorff dimension of this type of sets is difficult to be determined,even the box dimensions can be computed.
In this paper we shall give some estimations of the Hausdorff h-measure ofsome class of functions and we generalize the results from our previous paper([1]).
Definition. Let Rn be the n-dimensional Euclidean metric space.If r0 > 0 is a given number, then, a continuous function h(r) , defined on
[0, r0) , nondecreasing and such that limr→0
h(r) = 0 is called a measure function.
If 0 < δ <∞ , E is a subset of Rn and h is a measure function, the Hausdorffh-measure of E is defined by :
Hh(E) = limδ→0
inf
(Xi
h(|Ui|) : E ⊆[i
Ui : 0 < |Ui| < δ
).
where || denotes the diameter of a set.Particularly, when h(r) = rs, 0 < s < ∞, then the s-dimensional Hausdorff
measure of E, denoted by H s(E), is obtained.Let us define:
g(x) =
8>><>>:2x , 0 ≤ x < 1
2
−2 (x− 1) , 12≤ x < 3
2
2(x− 2) , 32≤ x < 2
(1)
f(x) =
∞Xi=1
λs−2i g(λix), (∀)x ∈ [0, 1] , (2)
where g is given in (1), s > 0, ε > 1 and λii∈N∗ is a sequence such that
λi+1 ≥ ελi > 0, (∀) i ∈ N∗. (3)
The following results will be proved:Theorem 1. If h is a measure function, h(t)˜tp, p ≥ 2, f is the function
defined in (2), with s ∈ (0 , 2) and λii∈N∗ ∈ R+ is a sequence that satisfies(3), then Hh(Γ(f)) < +∞.
Theorem 2. Let h be a measure function, such that
h(t)˜P (t)eT (t), t ≥ 0, (4)
105
where P and T are polynomials:
P (t) = a1t+ a2t2 + ...+ apt
p, p ≥ 1,
T (t) = b0 + b1t+ ...+ amtm,
with the propertyP ′ (t) + P (t) · T (t) > 0, t ≥ 0.
If f : [0, 1] → R is a δ - class Lipschitz function, δ ≥ 1, then: Hh(Γ(f)) < +∞.The result remains true if p ≥ 2, a1 = 0 and δ ∈ [0, 1] .Theorem 3. If Γ(f) is the graph of the function defined in (2) , s ∈ [0 , 2) ,
λii∈N∗ ∈ R+ is a sequence that satisfies (3) and h is a measure function satis-fying (1) , then Hh(Γ(f)) < +∞.
Theorem 4. If f : [0, 1] → R is a δ -class Lipschitz function, δ > 0 andh is a measure function such that h(t) ∼ ettp , p > 2 , then Hh(Γ(f)) = 0 . Theassertion remains true if p ≥ 1 and δ > 1.
Theorem 5. If E ⊂ Rn is a k-rectifiable set and h is a measure function suchthat h(t) ∼ ettp , p > 2 , then Hh(E) = 0 .
Theorem 6. Let A be any Hausdorff measurable set, with dimH A = p ,0 < p ≤ 1 on Ox axis and B any Lebesgue measurable set on Oy axis, such that0 < m1(B) < ∞ . If h is a measure function such that h(t) ∼ ettp+1 , then thereis a constant, c, such that
Hh(A×B) < c ·Hp(A) ·m1(B),
where m1 is the linear Lebesgue measure.
References
[1] A. Barbulescu, New results about the h-measure of a set, Analysis and Opti-mization of Differential Systems, Kluwer Academic Publishers, 2003, p. 43 -48.
[2] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics,Cambridge University, 1985
[3] K. J. Falconer, Fractal geometry: Mathematical foundations and applications,J.Wiley & Sons Ltd., 1990
[4] P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Pro-ceedings of Cambridge Phil. Soc., vol. 42, 1946, p. 15-23
”Ovidius” University of Constantza,Bd. Mamaia 124, 900527, Constantza, Romaniae-mail: [email protected]
106
Probabilistic approach to the construction ofweak solutions of nonlinear parabolic systems
Yana Belopolskaya, Russia
A probabilistic approach to the investigation of smooth solutions to systemsof nonlinear parabolic equations was developed by Yu.Dalecky and the author [1].Later we have applied this approach to construct smooth local in time solutions ofthe Cauchy problem for the Navier- Stokes system and the system of gas dynamicsequations. Here we discuss the way to extend this approach using the theoryof stochastic flows due to Kunita [2] and his approach to the construction oflinear parabolic equation weak solutions. Our main goal is to construct diffusionprocesses associated with weak solutions of the following Cauchy problem
∂uk∂t
= (a(x, u),∇)uk +1
2Fij(x, u)∇i∇ju+Bilk(x, u)∇iul + clk(x, u)ul,
uk(0, x) = u0k(x), k = 1, 2, . . . , d1, i, j = 1, . . . , d, (1)
where Fij =Pdk=1AikAjk and the summation over repeated indices is assumed.
We say that a C1,1 function u(t, x) is a weak solution to (1) providedZRd
vk(x)∂uk(t, x)
∂tdx =
ZRd
vk(x)[ai(x, u)∇iuk(t, x)+1
2Fij(x, u)∇i∇juk(t, x)]dx+
ZRd
vk(x)[Bilk(x, u)∇iul(t, x) + clk(x, u)ul(t, x)]dx, uk(0, x) = u0k(x) (2)
for any C1 smooth function v(x) with compact support.To construct a weak solution to (1) we consider a stochastic system
dψ(τ) = a(ψ(τ), u(τ, ψ(τ)))dτ +A(ψ(τ), u(τ, ψ(τ)))dw(τ), (3)
dη(τ) = c(ψ(τ), u(τ, ψ(τ)))η(τ)dτ + C(ψ(τ), u(τ, ψ(τ)))(η(τ), dw), (4)
〈h, u(t, x)〉 = E〈η(t), u0(ψt,0(x))〉, (5)
where w(τ) = w(t − τ) − w(t) and ψt(x) = x ∈ Rd, η(0) = h ∈ Rd1 and 〈·, ·〉 isthe inner product in Rd1
Note that the process ψ(τ) ≡ ψt,τ (x) may be treated as a process inverse toa stochastic process φ(τ) = φ0,τ (x) satisfying the SDE
dφ(τ) = −[a− Tr∇AA](φ(τ), u(τ, φ(τ)))dτ −A(φ(τ), u(τ, φ(τ)))dw(τ), φ(0) = x.(6)
Condition C1 . Let all coefficients and u0(x) be C1+α -smooth in x and in u .Assume as well that a and A have sublinear growth in x , c and Ch are boundedin x and all of them are of polynomial growth in u .
Theorem 1. Assume that C1 holds. Then there exists a unique solution(ξ(t), η(t), u(t, x)) of the system (3)-(5). The process ξ(t) is a Markov process,the process η(t) gives rise to a multiplicative operator functional S(t, s)h = η(t)
107
of ξ(t) and u(t, x) is a C1 smooth bounded function provided u0(x) is C1 andbounded.
Theorem 2. Under the conditions of theorem 1 the function u(t, x) given by(5) is a unique weak C1 -solution of the Cauchy problem (1).
To prove these results we have first to construct the solution to (3)-(5) withrequired properties and check that u(t, x) is a required weak solution to (1).
To construct a solution to (3)-(5) we consider a system of successive approxi-mations
u0(t, x) = u0(x), ψ0(t) = x, (7)
dψkt,τ (x) = a(ψkt,τ (x), u(τ, ψkt,τ (x)))dτ+ (8)
A(ψkt,τ (x), uk(τ, ψkt,τ (x)))dw(τ), ψkt,t(x) = x,
dηk(τ) = c(ψkτ,0(x), uk(ψkτ,0(x))ηk(τ)dτ+ (9)
C(ψkτ,0(x), uk(τ, ψkτ,0(x)))(ηk(τ), dw(τ)), ηk(0) = h,
〈h, uk+1(t, x)〉 = E〈ηk(t), u0(ψkt,0(x))〉. (10)
Notice, that at each step of the successive approximations scheme both equations(8) and (9) are well defined since the process ψk(τ) is adapted to both σ -algebraflows Fτ and Fτ generated by w(τ) and w(τ) respectively.
Actually, we can check that the process ψ(t) gives rise to a stochastic flowinverse to the flow generated by the solution of (6) and satisfies along with (3)the following SDE
dψτ,0(x) = ∇φ0τ (ψτ,0)−1a(x, u(τ, x))dτ−∇φ0,τ (ψτ,0)−1A(x, u(τ, x))dw(τ). (11)
To prove the uniform convergence of successive approximations uk(t, x) givenby (10) we add to (3)-(5) the equations for first derivatives of ξ(t), η(t) and u(t, x)in x and check the convergence of successive approximations of this extendedsystem solutions. Finally, the equation (11) and the generalized Ito formula allowto check that u(t, x) given by (5) is a weak solution to (1).
The support of Grant DFG 436 RUS 113/823 is gratefully acknowledged .
References
[1] Ya. Belopolskaya, Yu. Dalecky. Investigation of the Cauchy problem for qua-silinear parabolic systems with the help of Markov random processes. Izv,VUZ. Matematika N 12, 6: 1978.
[2] H. Kunita Stochastic flows and stochastic differential equations. CambridgeUniv. Press, Cambridge, 1990.
St.Petersburg State University for Architecture and Civil Engineering,Department of Mathematics,2-ja Krasnoarmejskaja 4, St.Petersburg, 190005, Russiae-mail: [email protected]
108
Arbitrage with fractional Brownian motion?Christian Bender, Germany
In recent years the modelling of financial markets via fractional Brownian motionsgained considerable interest. It has been extensively discussed whether it is possi-ble to set up an economically meaningful and arbitrage free fractional Brownianmotion pricing model (the no-arbitrage property meaning that there is no oppor-tunity for a riskless gain by trading into the market model). Clearly, existenceof an arbitrage crucially depends on the class of strategies which the investor isallowed to employ, and, of course on the exact specification of stock model. Inthis talk we will survey and discuss several results stating existence of arbitragerespectively absence of arbitrage in models driven by fractional Brownian motion.
A pure fractional model is driven by a fractional Brownian motion with Hurstparameter H > 1/2 without any Brownian motion component. One typicallydistinguishes between the ‘pathwise’ model and the ‘Wick’ model. Here ‘pathwise’means the model is based upon the ordinary product while ‘Wick’ means it is basedupon the Wick product. Comparing different approaches to pure fractional modelswe arrive at the following conclusions:
1. If the stock model and the self-financing condition is based on the ordinaryproduct, then there is an arbitrage or the class of allowed strategies is toosmall for hedging purposes.
2. The same holds, if the model is based on the Wick product, but the self-financing condition in given in terms of the ordinary product.
3. An arbitrage free class of strategies which is rich enough for hedging pur-poses exists, if both stock model and self-financing condition are based onthe Wick product. However, the Wick-self-financing condition has no cleareconomic interpretation.
It, hence, seems that there is no pure fractional model which is economicallymeaningful, arbitrage free and sufficiently rich for hedging purposes.
In the second part of the talk we will consider mixed fractional models, i.e.models that are driven by a fractional Brownian motion with H > 1/2 andby an independent Brownian motion. We shall only consider the economicallymeaningful case, when the self-financing condition is in terms of the ordinaryproduct. Under this self-financing condition, we will identify a class of strategieswhich is arbitrage free and rich enough to contain hedges for European options,and some important exotic options such as lookback options and Asian option. Wewill finally indicate, how the independence assumption can be relaxed and howthe no-arbitrage result can be extended to a broader class of no-semimartingalemodels.
Parts of the talk are based on joint works with R. J. Elliott, T. Sottinen andE. Valkeila.
Weierstrass Institute for Applied Analysis and Stochastics,Mohrenstr. 39, D-10117 Berlin, Germanye-mail: [email protected]
109
On the limit theorem for semi-Markov process.Ganna I. Bondarenko, Ukraine
A lot of papers are dedicated to asymptotic behavior of semi-Markov process ξ(t)as t→∞ . The techniques of inversion of perturbed on spectrum operators [1]and Tauber theorems [2] were used for investigation of the behavior of Markovrenewal function of semi-Markov process in [3]. On basis of this investigation itwas obtained in [3], [4] that if a strongly regular semi-Markov process ξ(t) withphase space X,B and semi-Markov kernel Q(t, x,B) , t ≥ 0 , x ∈ X , B ∈ Bsatisfies conditions C1 – C3, stated below, then there exists
Un(x,B) = limp→0
(−1)n
n!
∞R0
e−p ttnh∗(t, x,B)dt, n = 0, k, k ≥ 1, x ∈ X, B ∈ B,
where h∗(t, x,B) = h(t, x,B)−hc(B), h(t, x,B)— density of the Markov renewalfunction of the process ξ(t) , hc(B) — stationary density of the Markov renewalfunction. Note that generally speaking h∗(t, x,B) is not of constant signs on[0,∞) as x ∈ X, B ∈ B are fixed. Paper [4] gives recurrence relations which
relate limits in (??) to Ml(x,B) =∞R0
tlQ(dt, x,B), x ∈ X, B ∈ B, l = 0, k + 2
and to stationary projector Π0(B), B ∈ B of the embedded in ξ(t) Markov chain.In this report we present the following theorem, which is proved by means of
above mentioned results and Markov renewal theorem [5]:Theorem 1. If a strongly regular semi-Markov process ξ(t) satisfies conditions
C1. Markov chain ξn, n ≥ 0, embedded in the ξ(t) is uniformly recurring.C2. sup
x∈XMl(x,B) <∞, x ∈ X, B ∈ B, l = 1, k + 2, k ≥ 1 .
C3. Semi-Markov kernel of the process ξ(t) is absolutely continuous in t :
Q(t, x,B) =tR0
q(s, x,B)ds , t ≥ 0 , x ∈ X , B∈B,then
limt→∞
Un(t, x,B) = Un(x,B), n = 0, k, x ∈ X, B ∈ B, (1)
where U0(t, x,B) =tR0
h∗(t, x,B)dt, Un(t, x,B) =tR0
(Un−1(t, x,B)−Un−1(x,B) )dt,
n = 1, k. Note that truth of (1) in the case n = 0 follows from [3] but under twoadditional conditions.1. Korolyuk VS and Turbin AF . Mathematical Foundations for Phase Consolida-tion of Complex Systems.- Kyiv: Naukova Dumka, 1978 - 219 pp.2. Danford N., Schwarts JT. Linear Operators. General Theory. - M.: IL, 1962.3. Korlat AN, Kuznetsov VN, Novikov MM, and Turbin AF. Semi-Markov Modelsof Renewal and Queuing Systems.-Kishinev: Shtiintsa, 1991. - 276 pp.4. Bondarenko H.I. On Some Consequences of the Equation for the Markov Re-newal Function of a Semi-Markov Process. Ukrain.Math.J., 2004, Vol. 56, No. 12,pp. 1684 – 1690.5. Shurenkov. Ergodic Markov Processes. - Moscow: Nauka, 1989 - 336 pp.Shevchenko Kyiv National University, Kyiv, Ukrainee-mail: [email protected]
110
About the size of premium for an insurancecompany
Bondarev B.V., Smolyakov A.I., Stepanov E.V., Ukraine
We consider of insurance company when investments on (B,S)-market, wherec is size of premium, the stream of lawsuits is described by the difficult process ofPoisson with the parameter of λ , rate of bank is r. we used the cost of share fromthe model of Samuelson. Equation for a capital
dR(t) = [R(t)((1− u)r + uµ) + (c− aλ)− u1]dt+ uσR(t)dw −Z ∞
0
ueυ(du, dt)
Where u is a part of surplus, which is invested in share, u1 is a self-consumption.Functional of quality
V (t, x) = maxu,u1
M
Z T
t
e−ρsu1ds
Bellman equation
−∂V∂t
(t, x) = maxu,u1
(1
2u2x2σ2 ∂
2V
∂x2(t, x)+
Z ∞
0
(V (t, x+u)−V (t, x)−∂V∂x
(t, x)u)λF (du)+
+∂V
∂x(t, x)(x[(1− u)r + uµ] + (c− aλ)− u1) + e−ρtu1)
Boundary conditions
V (T, x) = 0,∂V
∂x(T, x) = 0
and0 ≤ u1(t, x) ≤ x[(1− u)r + uµ] + (c− aλ)
Let u1(t, x) meets condition Lipschitz. For insurance company we have: ifV (0, R(0)) = L then
C = aλ+Lρ
e−ρt0 − e−ρT−R(0)µ
wheret0 = T − 1
ρln(
µ
µ− ρ)
Institute of Applied Mathematics and Mechanicsof National Academy of Sci. of Ukraine,Roza Luxemburg st.74, Donetsk 83114, Ukraine;
Donetsk National University,Universitetskaya st.24, Donetsk 83055, Ukrainee-mail: [email protected]; [email protected]
111
Investigation of third order random oscillationsystem
Olga Borysenko, Oleksandr Borysenko, Ukraine
We study the behaviour, as ε→ 0 , of the third order autonomous oscillatingsystem described by stochastic differential equation
x′′′(t) + ax′′(t) + b2x′(t) + ab2x(t) = εk1f1(x(t), x′(t), x′′(t)) + fε(x(t), x′(t), x′′(t))(1)
with non-random initial conditions x(0) = x0, x′(0) = x′0, x
′′(0) = x′′0 , whereε > 0 is a small parameter, fε(x, x′, x′′) is a random function such thatZ t
0
fε(x(s), x′(s), x′′(s)) ds = εk2Z t
0
f2(x(s), x′(s), x′′(s)) dw(s)+
+εk3Z t
0
Zf3(x(s), x′(s), x′′(s), z) ν(ds, dz),
ki > 0, i = 1, 2, 3 ; fi, i = 1, 2, 3 are non-random functions; w(t) is a standardWiener process; ν(dt, dy) = ν(dt, dy)−Π(dy)dt , Eν(dt, dy) = Π(dy)dt , ν(dt, dy)is the Poisson measure independent on w(t) ; Π(A) is a finite measure on Borelsets in R .
It is natural, that the equation (1) we understand as system of stochasticdifferential equationsdx(t) = x′(t)dt ,dx′(t) = x′′(t)dt ,dx′′(t) = [−ax′′(t)− b2x′(t)− ab2x(t) + εk1f1(x(t), x′(t), x′′(t))]dt++εk2f2(x(t), x′(t), x′′(t))dw(t) + εk3
Rf3(x(t), x′(t), x′′(t), z)ν(dt, dz) ,
x(0) = x0, x′(0) = x′0, x
′′(0) = x′′0 .Let
x(t) = C(t) +A(t) cos(bt+ θ(t)),
x′(t) = −aC(t)−A(t)b sin(bt+ θ(t)),
x′′(t) = a2C(t)−A(t)b2 cos(bt+ θ(t)),
k = min(k1, 2k2, 2k3) . We obtained the sufficient conditions for weak convergenceof the stochastic process ξε(t) = (C(t/εk), A(t/εk), θ(t/εk)) , as ε → 0 , to thesolution ξ(t) = (C(t), A(t), θ(t)) of the averaging system of stochastic differentialequations
dξ(t) = α(ξ(t))dt+ σ(ξ(t))dw(t),
where w(t) – is some Wiener process. The dependence of coefficients of averagingsystem on order of small parameter ε > 0 is investigated.
Department of Mathematical Physics,National Technical University of Ukraine (KPI), Kyiv, Ukraine.Department of Probability Theory and Mathematical Statistics,Kyiv National Taras Shevchenko University, Kyiv, Ukrainee-mail: [email protected]
112
Iterated logarithm law for solutions ofstochastic equations
Dmitry Budkov, Ukraine
Let (Ω,F , P ) – be the probability space. The space C[0, 1] – is the space ofall continuous functions with domain [0, 1] taking values in R1.
Define Φ – the class of increasing functions ϕ(h), h ≥ 0 such that
limh→0
ϕ(h) = 0, limh→0
√h
ϕ(h)= 0.
For every ϕ ∈ Φ define R2(ϕ) by the rule
R2(ϕ) = inf
(r > 0 :
∞Xn=1
exp
−rϕ
2(cn)
2 cn
ff<∞
),
If there does not exist an r <∞, to converge the series, we consider R2(ϕ) = ∞.Let us remark that if series converges for the certain c0 ∈ [0, 1], this will be truefor every c ∈ [0, 1].
Fixing the functional I0(f) = 12
R 1
0[f ′(t)]2dt define classes of functions
KR = f ∈ C[0, 1] : f(0) = 0, I0(f) < R2/2,
H =nZ t
0
g(s) ds : g(t) = σ(0)f(t), f ∈ KR, t ∈ [0, 1]o.
If R = ∞, then KR = f ∈ C[0, 1] : f(0) = 0.Let x(t) be the solution of stochastic equation
x(t) =
Z t
0
b(x(s))ds+
Z t
0
σ(x(s))dw(s). (1)
Theorem 1 Let the coefficients of (1) b(x) and σ(x) are bounded, uniformlycontinuous, diffusion σ(x) is Lipschitz and σ(0) 6= 0, ϕ ∈ Φ. Then the sequenceas h→ 0
1h
R th0x(s) ds
ϕ(h), t ∈ [0, 1],
has almost surely limit set equal H in C[0, 1]. Upper and lower limits are accord-
ingly ± R
σ(0)√
3.
Institute of Applied Mathematics and Mechanics NAS of Ukraine,R.Luxembrg street, 74, Donetske-mail: [email protected].
113
Large deviation, regularly varying functions,and precise asymptotics over a small
parameter of a generalized Spitzer seriesV. V. Buldygin, Ukraine; O. I. Klesov, Ukraine; J. G. Steinebach, Germany
There are three topics combined in this talk. The first of them is the largedeviation principle for sums of independent identically distributed random vari-ables attracted to a stable law of index α , α < 2 . Our result has its roots in atheorem by Heyde (1968) who obtained a similar result in the case of attractionto a nondegenerate α -stable law if α 6= 1 and α 6= 2 , i.e.
limn→∞
P(|Sn| ≥ xnbn)
nP(|X| ≥ xnbn)= 1
for any sequence xn such that xn → ∞ as n → ∞ . For the obvious reasonHeyde’s result can be called the asymptotic large deviation result. Here and inwhat follows bn means the normalizing sequence in the weak convergence tothe limiting stable law. In contrast we prove the uniform large deviation result.
Theorem 1. Assume that 0 < α < 2 . Let X , Xn be a sequence of i.i.d.random variables such that X belongs to the domain of attraction of a stable lawof index α and with normalizing sequence bn . If α = 1 we assume that X hasa symmetric distribution, while if α > 1 let EX = 0 . Then
supx≥0
supn≥1
P(|Sn| ≥ xbn)
nP(|X| ≥ xbn)<∞.
The other topic is the asymptotic behavior of integrals and sums of regularlyvarying functions. One of challenging results for slowly varying functions is due toParameswaran (1961) saying that if w is a slowly varying function and r = −1 ,then W is a slowly varying function such that wn = o(Wn) where
Wn =
nXk=1
krwk. (1)
The case of r > −1 is due to Karamata and is absolutely different. We use theabove result to determine the asymptotic behavior of
Q(ε) =
∞Xk=1
wkP(|Z| ≥ εψk), ε ↓ 0,
for a given random variable Z and regularly varying functions w and ψ . Theearlier results concerning this series are known only for stable random variablesZ . Our assumption is that
E|Z|η <∞ for some η >r + 1
γand for some η <
r + 1
γ.
114
This condition for r 6= −1 is superfluous; but for r = −1 every its part has aspecial value. The earlier methods of proof make use of Euler–Maclaurin summa-tion formula which nicely works only for power functions. Our method is muchmore general and based on some asymptotic results of Aljancic, Bojanic, andTomic (1954).
The third topic unifying the previous two ones is the asymptotic behavior ofgeneralized Spitzer series. This topic has an extended literature (see, for example,Spitzer (1956), Heyde and Rohatgi (1967), Chow and Lai (1978), Chen (1978),Spataru (1999), Gut and Spataru (2000), Scheffler (2003), Rozovskii (2003), Gutand Steinebach (2004) for particular cases) and deals with the series
Q(ε) =
∞Xk=1
wkP (|Sk| ≥ εϕk) (2)
where Sk are partial sums of independent identically distributed random vari-ables of the domain of attraction of a stable law; w and ϕ are regularly varyingfunctions. We express the asymptotic behavior of the generalized Spitzer series interms of functions w and ϕ and normalizing sequences in the attraction to thestable law.Theorem 2. Let X and Xn be independent identically distributed randomvariables. Assume that X is attracted to a stable law of index α . In addition,we assume that the centering sequence vanishes if α = 1 or EX = 0 if α > 1 .Denote by Zα an α -stable random variable to which X is attracted. Let w and ϕbe regularly varying functions with indices r and 1/p , respectively, and such thatr ≥ −1 and 0 < p < α . Let U(x) = W
`ψ−1(x)
´where W is defined by (1) and
ψ(x) = ϕ(x)b(x)
, b is the normalizing function in the attraction to Zα . If the seriesPwk diverges and α > p(r + 2) , then
limε↓0
Q(ε)
U(1/ε)= E|Zα|(r+1)αp/(α−p)
where Q is defined by (2).For example, if random variables Xn are attracted to the Gaussian limit,
then the asymptotics of the original Spitzer series is as follows
limε↓0
1
− ln(ε)
∞Xn=1
1
nP(|Sn| ≥ nε) = 2.
Department of Mathematical Analysis and Probability Theory,National Technical University of Ukraine (KPI),pr. Peremogy, 37, Kyiv 03056, Ukrainee-mail: [email protected] ; [email protected]
Universitat zu Koln, Mathematisches Institut,Weyertal 86–90, D–50931 Koln, Germanye-mail: [email protected]
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Limit Theorems for Associated and RelatedRandom FieldsAlexander Bulinski, Russia
There are a number of important stochastic models described by positively ornegatively associated random fields. The main sources of interest here are stud-ies in statistical physics, percolation theory, mathematical statistics and reliabil-ity theory after the pioneering papers by Harris, Lehmann, Esary, Proschan andWalkup, Fortuin, Kasteleyn and Ginibre, Simon, Lebowitz, Newman, Cox andGrimmett.
The lecture provides, in anticipation of the forthcoming book [1] by A.Bulinskiand A.Shashkin, not only a survey of achievements and trends in this researchdomain but includes recent results of the authors till 2006. The main topics arethe following.
1. Stochastic models and covariance inequalities.2. Moment and maximal inequalities for multiindexed summands.3. Central limit theorem.4. Law of large numbers.5. Weak invariance principle.6. Law of the iterated logarithm.7. Strong invariance principle.8. PDE with random initial data.9. Statistical applications.
By definition upgoing to Esary et al. (1967) real-valued random variablesX1, . . . , Xn are associated, or a vector X = (X1, . . . , Xn) is associated, if, forany bounded coordinate-wise nondecreasing Borel functions f : Rn → R andg : Rn → R , one has
cov(f(X1, . . . , Xn), g(X1, . . . , Xn)) ≥ 0. (1)
An infinite family Xt, t ∈ T is called associated whenever each its finite sub-family has the mentioned property. Note that any family of independent randomvariables is automatically associated. Due to Pitt (1982) a Gaussian process (field)Xt, t ∈ T is associated if and only if cov(Xs, Xt) ≥ 0 for any s, t ∈ T . Lee,Rachev and Samorodnitsky (1990) provide necessary and sufficient conditions forassociation of a stable random vector. Due to Evans (1990) any infinitely divisiblerandom measure on a Polish space is associated. There are many other usefulexamples.
Starting from the seminal paper by Newman (1980) during the last 25 yearsthe principle limit theorems were established for associated stochastic processesand random fields such as CLT, SLLN, LIL, weak and strong invariance principles,functional law of the iterated logarithm, Glivenko – Cantelli theorem, etc.
There are various modifications of the association property. For instance thenegative association introduced by Joag-Dev and Proschan (1983) means that
cov(f(XI), g(XJ)) ≤ 0 (2)
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for all bounded coordinate-wise nondecreasing Borel functions f : R|I| → R ,g : R|J| → R where |I| , |J | stand for the cardinalities of disjoint finite setsI, J ⊂ T and XI = (Xt, t ∈ I) , XJ = (Xt, t ∈ J) .
The changing of the sign of inequality (2) leads to the definition of positive (orweak) association introduced by Newman (1984). There are also other modifica-tions of (1). Thus if EX2
t <∞ , t ∈ T , and f, g are bounded Lipschitz functionsthen for positively or negatively associated family Xt, t ∈ T one has accordingto [2]
|cov(f(XI), g(XJ))| ≤ Lip(f)Lip(g)Xi∈I
Xj∈J
|cov(Xi, Xj)| (3)
where Lip(f) and Lip(g) are the Lipschits constants for f and g respectively.Thus it is natural to describe the dependence properties of a family Xt, t ∈ Tin termes of upper bounds for a functional
F (f, g; I, J) = |cov(f(XI), g(XJ))|, (4)
here f, g belong to a specified class of "test" functions (e.g., the Lipschitz ones)and I, J are disjoint finite subsets of T . An analogous approach was employedin the theory of mixing stochastic processes and random fields. However, the es-timation of mixing coefficients is much more involved than analysis using thehypotheses of positive (or negative) dependence and inequalities of the type (3)which allows to apply the well-known Cox-Grimmett coefficient. Moreover, theconditions of limit theorems take a unified form. Namely, one supposes the ex-istence of the moment E|Xt|s for some s ≥ 2 and the decreasing rate of theCox-Grimmett coiefficient (or related coefficients). During the last decade suchapproach to dependence of stochastic processes and random fields based on theestimates of F appearing in () was developed in the papers by Doukhan andLouhichi [3], Bulinski and Suquet [4] and other researchers.
[1] A.Bulinski, A.Shashkin. Limit Theorems for Associated and Related RandomFields. 2006 (to appear), 350 pp.[2] A.Bulinski, E. Shabanovich, Asymptotical behaviour for some functionals ofpositively and negatively dependent random fields. Fund. Prikl. Mat., 1998, 4, No2, p. 479-492 (in Russian).[3] P.Doukhan, S.Louhichi. A new weak dependence condition and application tomoment inequalities. Stoch. Proc. Appl., 1999, 84, No 2, p. 313-342.[4] A.Bulinski, Ch.Suquet. Normal approximation for quasi-associated randomfields. Statist. Probab. Lett., 2001, 54, No 2, p. 215-226.
Dept. of Mathematics and Mechanics,Moscow State University,Moscow 119992, Russia
e-mail: [email protected]
117
Normal asymptotic for continuous procedurestochastic approximation in semi-Markov space
Yaroslav Chabanyuk, Ukraine
Introduction. The procedures stochastic approximation(PSA) are used indifferent areas of applying mathematic statistic, management theory and passinginformation, recognition of the images theory, etcetera. Near the main problemof converges PSA there exists an important problem of the investigation the fluc-tuations PSA around the radical of the regression equation. The investigation ofthe asymptotic normal of the fluctuation in the classic schemes of PSA is realizedwith the use of the principle of inversion for the sums (in the descried scheme)and processes (in the continuous scheme) [1,2]. In our previous work [3] the as-ymptotic normality PSA in the Markov’s space is determined with the applianceof martingale characterization corresponding two-componential Markov’s processand using the solution of the problem of singular perturbation for the creation ofthe generator of the limited diffusion process of the Ornshtein-Uhlenbeck type.
The definition of the problem. The continuous PSA with semi-Markov’sswitching in the series scheme is set by evolution equation
duε(t)/dt = a(t)C(uε(t), x(t/ε2)) (1)on the real axis R = (−∞; +∞) . The regression function C(u, x) , u ∈ R , x ∈ X ,satisfies the condition of existing of global solution of the accompanying systems
dux(t)/dt = C(ux(t), x) , x ∈ X .The semi-Markov’s process of the switching x(t), t ≥ 0 , in the standard phase
space ( X,X ) is set by semi-Markov’s kernel [3] Q(x,B, t) = P (x,B) Gx(t) ,x ∈ X , B ∈ X , t ≥ 0 . The stochastic kernel P (x,B) sets the transitionalprobability invested Markov’s chain xn, n ≥ 0 ,P (x,B) = P xn+1 ∈ B|xn = x , and the distribution function Gx(t) , x ∈ X ,t ≥ 0 , define the moments of the Markov’s renewal τn+1 = τn + θn+1 , n ≥ 0 ,τ0 = 0 , Gx(t) = P θn+1 ≤ t|xn = x , x ∈ X , t ≥ 0 . Under the certainconditions on the normal function a(t), t ≥ 0, and uniform ergodic of the semi-Markov’s process x(t), t ≥ 0 , with the stationary distribution π(B), B ∈ X ,PSA, which defines by the solution of the evolutional equation (1), converges withprobability one to the equilibrium point u0 of the averaged system [3]
du(t)/dt = C(u(t)) , C(u0) = 0 , (2)with the regression function C(u) =
RXπ(dx)C(u, x) .
From now on not to decrease the generality, we consider u0 = 0 , so the conditionis taking place C(u) =0.
Notice. Converge PSA uε(t) ⇒ 0 , t → ∞ , it means, that it is expedient tostudy fluctuations with the next normalizing (see [3]): νε(t) =
√tuε(t)/ε .
Now we’ll examine the standard PSA with the normalizing functiona(t) = a/t , a > 0, t > 0.
Additional conditions on the regression function are the same as in the work[3], concretely: C(u, ·) ∈ C2(R) , the second derivative on u C′′u (u, ·) satisfies
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the global condition of Livshitch: |C′′u (u, x) - C′′u (u′, x)| ≤ C |u− u′| , with theconstant C , which don’t depend on x ∈ X .
It’s known, that uniform ergodic of the semi-Markov’s process x(t), t ≥ 0 ,with the semi-Markov’s kernel defines the generator Q : Qϕ(x) = q(x)RXP (x, dy)[ϕ(y)− ϕ(x)] , where q(x) := 1/g(x) , g(x) :=
R∞0Gx(t)dt , Gx(t) :=
1- Gx(t) , accompanying uniformly ergodic Markov’s process x0(t), t ≥ 0 . Herethe generator Q is reducibly invertible [3], for which the potential operator R0 ,which defines by the equations [3], exists. R0Q = QR0 = I −Π.
Here the projector Π in the Banah’s space B(X) real meaning functions withthe supreme norm is set by the relation:
Πϕ(x) :=_ϕ 1 (x) ,
_ϕ :=
RXπ(dx)ϕ(x) , 1 (x) ≡ 1 , x ∈ X ,
or the projector Π is projecting on the null-space of the generator Q .Taking into consideration the condition, we’ll use the formula of Taylor for
the regression functionC(u, x) = C0(x) + u C1(x) + u2
2C2(u, x) ,
Here according to the definitionC0(x) := C(0, x) , C1(x) := C′u(u, x)|u=0 , C2(u, x) := C′′u (θu, x) , 0 ≤ θ ≤ 1 .Let us put in necessary notes c := -
RXπ(dx)C1(x) , b := ac−1/2 , and another
moments of the times of being g2(x) :=R∞0t2Gx(t)dt = 2
R∞0G
(2)x (t)dt , where
G(2)x (t) :=
R∞tGx(s)ds . Let it be fulfilled the conditions of converge PSA (1) in
the semi-Markov’s space [3].Theorem (Asymptotic normality) . In the conditions of converge PSA (1)
in the semi-Markov’s space and with the additional conditionsA1: ρ := 2
RXπ(dx) [ C0(x) R0 C0(x) + µ(x) C2
0 (x)] > 0,where µ(x) := ( g2(x) -2 g2(x))/2g(x) ;
A2: b := ac− 1/2 > 0,the weak converge is taking place νε(t) ⇒ ζ(t), ε → 0, in every finite interval0 < t0 ≤ t ≤ T . The process ζ(t), t ≥ 0, is the Ornshtein-Uhlenbeck ’s diffusionprocess, which defines by generator L ϕ(v) = - b v ϕ′(v) + 1
2ρ ϕ′′(v) .
Conclusion. In the conditions of the theorem PSA νε(t) have asymptoticallynormal distribution N(0, σ2
0) , that is νε(t) ⇒ ν, ε → 0, t → ∞ . Accidentalquantity ν ∈ N(0, σ2
0) .
1. Nevelson M.B., Khasminskii R.Z. Stochastic approximation and recurrencevalue // Moscow: Nauka, 1972. - 304 p.(in Rusian).
2. Ljung L., Pflug G., Walk H.. Stochastic Approximation and Optimizationof Random Systems // Birkhauser Verlag, Basel, Boston, Berlin, 1992. -113 p.
3. Chabanyuk Ya.M. Asymptotic normality for the continuous procedure ofstochastic approximation in the Markov’s space // Dop. Nat. Academy NaukUkraine, 2004, s. А, 5, p. 37-45. (in Ukrainian)
National University "Lvivska Politechnika",Lviv, S.Banderi st.,12e-mail: [email protected]
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Exact non-ruin probabilities in infinite time
Chernecky V. A., Ukraine
Let F (v) be the distribution of claims Zk , EZk = µ , K(v) be the distributionof waiting time Tk , ETk = 1/α , k ∈ Z+ , and c > αµ be the gross premium rate.Probability of solvency of an insurance company, ϕ(u) , with initial capital u , inordinary renewal process, satisfies the integral equation, [1]:
ϕ(u) =
Z ∞
0
dK(v)
Z u+cv
0
ϕ(u+ cv − z) dF (z), ϕ(∞) = 1. (1)
In stationary renewal process, the probability of solvency, ϕ0(u) , is calculated bythe formula:
ϕ0(u) = α
Z ∞
0
[1−K(v)]
Z u+cv
0
ϕ(u+ cv − z) dF (z)
ffdv, ϕ0(0) = 1− αµ/c.
(2)Reducing the equation (1) by the difference method to a system of linear
algebraic equations, we notice that the received system is a discrete system ofWiener-Hopf type [2]. This makes us think that the equation (1) is a homogeneousintegral equation of Wiener-Hopf type
ϕ(u) =
Z ∞
0
ϕ(u− v) dk(v), 0 ≤ v <∞,
with some kernel k(v) depending on K(v) and F (v) . Using the technique devel-oped in the monographs [2,3], for the symbol A(λ) of (1) we receive the formula
A(λ) = 1−FT (cλ) · FZ(λ), −∞ < λ <∞,
where F·(·) denotes a characteristic function (Fourier transform) of a correspond-ing distribution. The symbol A(λ) is differentiable on the numerical line and hasalways the zero of the first order at λ = 0 . The latter follows from the conditionc > αµ . There is yet the possibility of existence of infinite number of other zerosλk of A(λ) on the numerical line. Therefore the equation (1) is of nonelliptic type,for the solution of which we can apply the Wiener-Hopf factorization method [2,3].There exists a factorization
A(λ) = λ · A+(λ) · A−(λ) ·Yk
(λ− λk),
where the functions A+(λ) (A−(λ)) assumed to be analytically extendable inthe domains Imλ > 0 (Imλ < 0) and continuous in (Imλ ≥ 0) (Imλ ≤ 0) . Thesolution of the problem (1) is represented in the form of a linear combination:
ϕ(u) = C1 F−1
»1
λ · A+(λ)
–+ C2 F−1
»1
A+(λ)
–= C1 ϕ1(u) + C2 ϕ2(u),
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where F−1[·] denotes the inverse Fourier transform in the sense of the L. Schwartzdistributions. Note that ϕ2 can be represented in the form of derivative ϕ′1 in thesense of L. Schwartz. The constants C1 and C2 are uniquely determined by thetwo conditions:
limu→∞
ϕ(u) = 1 and limu→+0
ϕ0(u) = 1− αµ/c.
Since the characteristic function for the gamma distribution with the positiveinteger parameters is a rational one, in the case when Zk and Tk have such dis-tributions we can always obtain the solution of (1) in the explicit form. Examplesare considered.
In the monograph [4], A. Melnikov investigated so called compound binomialmodel where Zk are independent identically distributed random variables withvalues in the set of all natural number N , the distribution of Tk ≥ 1 can be con-sidered as the ‘shifted’ geometrical distribution and c = 1 . A. Melnikov succeededin receiving an exact solution for this discrete model in the form of a generat-ing function Gϕ(z) for ϕ(u) , u ∈ Z+ . Assuming c > αµ to be an arbitrarypositive integer, we consider the problem in general setting, when Tk has an arbi-trary positive integer-valued distribution. Reducing the equation (1) to a discreteWiener-Hopf equation [2], for the symbol A(z) of which we obtain the expression
A(z) = 1−GT (z−c) ·GZ(z), |z| = 1,
where G·(·) denotes a corresponding generating function. The symbol A(z) hasalways the zero of the first order at z = 1 , and can have the finite number of otherzeros zk , |zk| = 1 , zk 6= 1 . Therefore the equation (1) in the discrete case also isof nonelliptic type, for the solution of which it is possible to use the Wiener-Hopffactorization method on the circle |z| = 1 , [2]. Repeating the above reasoningwith some modification, we obtain the unique solution of (1) in the form of thegenerating function for ϕ(u) .
Examples when the random values Tk and Zk be ‘shifted’ uniform discrete,binomial, geometrical, Poisson, negative binomial, are considered.
References
1. J. Grandell, Aspects of Risk Theory, Springer-Verlag, 1991.2. S. Prosdorf, Einige Klassen singularer Gleichungen, Akademie-Verlag-Berlin, 1974.3. F. D. Gakhov and Yu. I. Cherski, Equations of convolution type, (Russian)Nauka, Moskow, 1978.4. A. Melnikov, Risk analysis in finance and insurance, Chapman & Hall/CRC,2003.
Mechnikov Odessa National University,Department of Mathematical Analysise-mail: [email protected]
121
Approximate non-ruin probabilities in discretemodel
Chernecky V. A., Pokas’ S. M., Ukraine
Let F (x) be the distribution of claims Zk , EZk = µ , K(x) be the distributionof waiting time Tk , ETk = 1/α , k ∈ Z+ , and c > αµ be the gross risk premiumrate.
Probability of solvency of an insurance company, ϕ(u) , with initial capital u ,in ordinary renewal process, satisfies the integral equation
ϕ(u) =
Z ∞
0
dK(v)
Z u+cv
0
ϕ(u+ cv − z) dF (z), ϕ(∞) = 1. (1)
In stationary renewal process, the probability of solvency, ϕ0(u) , is calculated bythe formula
ϕ0(u) = ϕ0(0) +α
c
Z u
0
ϕ(u− z)[1− F (z)] dz, ϕ0(0) = 1− αµ
c, (2)
where ϕ(u) is the solution of (1), [1].In the monograph [2], by using the algebraic properties of the Toeplitz matri-
ces, A. Melnikov investigated so called compound binomial model where Zk areindependent identically distributed random variables with values in the set of allnatural number N , the distribution of Tk ≥ 1 can be considered as the ‘shifted’geometrical distribution and c = 1 . A. Melnikov succeeded in receiving an exactsolution for this discrete model in the form of a generating function Gϕ(z) forϕ(u) , u ∈ Z+ . Then ϕ(u) can be calculated, for example, by the function taylor
of the pack Maple-V.We consider the problem in the general setting, when Tk and Zk are the
arbitrary integer-valued random variables, c > αµ is not necessarily integer, andlook for the corresponding approximate solution of the equation (1).
Let us introduce the system of nodes on some sufficiently wide interval,
uj = j, j = 0, 1, . . . , N.
and reduce the equation (1) by the difference method to a system of linear homo-geneous algebraic equations,
A eϕ = 0, (3)
with a (N + 1)× 2 (N + 1) Toeplitz matrix A and the vector of unknowns:
eϕ = (ϕ0, ϕ1, . . . , ϕN , ϕN+1, . . . , ϕ2N+1)′.
Replacing the initial condition ϕ(∞) = 1 in (1) by the close one:
ϕk = 1, k = N + 1, . . . , 2N + 1, (4)
122
for an approximate solution bϕ , the integral equation (1) is reduced to a systemof nonhomogeneous linear algebraic equations with a (N + 1)× (N + 1) Toeplitzcoefficient matrix of large dimension (of order of several thousands),
A0 bϕ = y, (5)
with a vector of unknowns bϕ = (ϕ0, ϕ1, . . . , ϕN )′, and with some right part y .For the solution of the system (5), a program on the programming language C iscomposed. The program uses a subroutine toeplitz(r,x,y,n+1) from [4] on thefast solution of linear algebraic systems with Toeplitz matrix. The approximatesolution, eϕ0 , in the stationary renewal process, is obtained by the correspondingdiscrete analog of the formula (2).
The exact solutions in the discrete model is known us for such distributionsof Zk and Tk as binomial, Poisson, geometrical, negative binomial. Note that forthe random variables Zk these distributions mean the shifted/conditional distri-butions at the condition that Zk ≥ 1 .
Comparing the exact solutions with approximate ones, we observe that thesesolutions coincide up to 13-15 decimal signs after decimal point. This demonstratesthe effectiveness of the method.
As if we use the approximate initial condition (4), the maximal error of approx-imate solutions is observed at the point u = 0 . To estimate the error in generalcase, we use the approximate solution in stationary model since for this solutionis known the initial condition at u = 0 . In general, the approximate solution of(2) can tend not to 1 when u→∞ . Therefore we normalize this solution by thefollowing way: eϕ2N+1 = 1 . Then the normalized solution satisfies not the initialcondition in (2). The divergence of these two values, exact and approximate inu = 0 , allows to judge about the exactness of the approximate solution (1).
The distributions considered above are light-tailed ones. It is interesting toconsider the heavy-tailed distributions. To this end we consider the discretizedheavy-tailed distributions: lognormal, Pareto, Burr, Benktanter-type I-II, Weibull,loggamma (shifted or conditional at the condition that Zk > 0).
References
1. J. Grandell, Aspects of Risk Theory, Springer-Verlag, 1991.2. A. Melnikov, Risk analysis in finance and insurance, Chapman & Hall/CRC,
2003.3. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numericalrecipes in C: The art of scientific computing. Second edition, Cambridge UniversityPress, 1988-1992.
Mechnikov National Odessa University,Department of Mathematical Analysis,Department of Geometry and Topologye-mail: [email protected]; [email protected]
123
Stochastic calculus and Arratia flow
Dorogovtsev Andrey A., Ukraine
The main object of the talk is the Arratia flow of coalescing Brownian particlesx(u, t);u ∈ R, t ≥ 0 [1 – 3]. The stochastic calculus for this flow is proposed. Inparticular the new type of the stochastic integral with respect to x is introduced.This allows to obtain the Girsanov theorem for coalescing flows, Clark represen-tation theorem for the functionals from x. The Markov process in the space ofmeasurable mappings on R related to x also is investigated. The certain posi-tive additive functionals from this process are described in the terms of specialmeasures on R.
References
1. Arratia, R.A. (1979). Brownian motion on the line. PhD dissertation Univ.Wiskonsin, Madison.
2. Le Jan, Yves, Raimond, Oliver. (2004). Flows, coalescence and noise. TheAnnals of Probability, Vol.32, No. 2, 1247 – 1315.
3. Dorogovtsev, A.A. (2005). Some remarks on the Wiener flow with coales-cence. Ukrainian Math. Journ. Vol. 57, No. 10, 1327 – 1333.
Institute of Mathematics Ukrainian ASe-mail: [email protected]
124
Weak dependence, models and applicationsPaul Doukhan, France
For a long time mixing conditions have been the dominating type of condi-tions for imposing a restriction on the dependence between time series data. Theyare considered to be useful since they are fulfilled for many classes of processesand since they allow to derive tools similar to those in the independent case. Onthe other hand, it turns out that certain classes of processes which are of interestin statistics are not mixing although a successive decline of the influence of paststates takes place. The simplest example of such a process is an AR(1)-process,Xt = 1
2Xt−1 + εt , where the innovations are independent and identically distrib-
uted with P (εt = 1) = P (εt = −1) = 1/2 , see [5]. In statistics (e.g for bootstrap)such discrete inputs naturally arise.Definition 1 ([2]) A process (Xt)t∈Z is called weakly dependent if there exists auniversal null sequence (εr)r∈N such that, for any k -tuple (s1, . . . , sk) and anyl -tuple (t1, . . . , tl) with s1 ≤ . . . ≤ sk < sk + r = t1 ≤ . . . ≤ tl and arbitrarymeasurable functions g : Rk → R , h : Rl → R with ‖f‖∞ ≤ 1 and ‖g‖∞ ≤ 1 ,the following inequality is fulfilled:
|cov (h(Xs1 , . . . , Xsk ), g(Xt1 , . . . , Xtl))| ≤ ψ(k, l,Lip f,Lip g) εr.
Here Liph denotes the Lipschitz modulus of continuity of h , that is,
Liph = supx6=y
|h(x)− h(y)|‖x− y‖l1
, where ‖z‖l1 =Xi
|zi|,
and ψ : N2 ×R2+ → [0,∞) is an appropriate function.
- κ -weak dependence for which ψ(u, v, a, b) = uvab , in this case we simply denoteεr as κr ,- κ′ (causal) weak dependence for which ψ(u, v, a, b) = vab , in this case we simplydenote εr as κ′r ;- η -weak dependence, ψ(u, v, a, b) = ua+ vb , in this case we write εr = ηr ,- θ -weak dependence is a causal dependence which refers to ψ(u, v, a, b) = vb , wesimply denote εr = θr ([1]) for this causal counterpart of η coefficients,- λ -weak dependence ψ(u, v, a, b) = uvab+ua+ vb , in this case we write εr = λr([4]). Besides the fact that it includes η - and κ -weak dependence, this new no-tion of λ -weak dependence is convenient, for example, for Bernoulli shifts withassociated inputs.Examples(see [2] and [4]).- Gaussian or associated processes with limt→∞ |cov (X0, Xt)| = 0 , are κ -weaklydependent with κr = O
`supt≥r |cov (X0, Xt)|
´or κ′r = O
“Pt≥r |cov (X0, Xt)|
”.
- ARMA(p, q) processes and more generally causal or non-causal linear processes:Xt =
P∞k=−∞ akξt−k , if ak = O(|k|−µ) with µ > 1/2 , then ηr = O
`1
rµ−1/2
´. For
dependent inputs see [4].- LARCH(∞) (see [3]): Xt = ξt
`b0 +
P∞k=1 bkXt−k
´(include GARCH(p, q)
processes)with ‖ξ0‖m ·P∞j=1 |bj | < 1 , if bj = 0 for j > J then θr = O(e−cr) ,
|bj | ≤ C · µ−j , then θr = O(e−c√r) , and |bk| ≤ C · k−ν , then θr = O
`r−ν+1
´.
125
For Non-causal processes Xt = ξt`b0 +
Pk 6=0 bkXt−k
´coefficients η replace θ if
‖ξ0‖∞ ·P∞j=1 |bj | < 1 .
- Very general models are the causal or non-causal infinite memory processesX = (Xt)t∈Z such that Xt = F (Xt−1, Xt−2, . . . ; ξt), and Xt = F (Xs, s 6= t; ξt),where the functions F defined either on RN\0 × R or RZ\0 × R satisfy‖F (0; ξ0)‖m < ∞, and ‖F ((xj)j ; ξ0) − F ((yj)j ; ξ0)‖m ≤
Pj 6=0 aj |xj − yj | with
a =Pj 6=0 aj < 1 . Then, works in progress by Doukhan and Wintenberger as well
as Doukhan and Truquet, respectively, prove that a solution of the previous equa-tions is stationary in Lm and either θ -weakly dependent or η -weakly dependentwith the decay rate: infp≥1
nar/p +
P|j|>p aj
o. This provides the sames rates
that those already mentioned for the case of LARCH(∞) models.Central limit theoremsTheorem 1 (Donsker type results)Assume that the zero mean stationaryprocess (Xt)t∈Z satisfies E|X0|m < ∞ , for m => 2 . Then the previous expres-sion σ2 =
Pt∈Z EX0Xt ≥ 0 is well defined and the Donsker invariance principle
Wn(t) = 1√n
Pnti=1Xi →n→∞ σW (t), in distribution to a Brownian motion in
the Skorohod space D([0, 1]), holds if one of the following additional assumptionsholds:κ-dependence, κr = O(r−κ) (as r ↑ ∞) for some κ > 2 + 2/(m− 2) ,λ-dependence, λr = O(r−λ) (as r ↑ ∞) for λ > 4 + 2/(m− 2) ,θ -dependence, θr = O(r−θ) (as r ↑ ∞) for θ > 1 + 1/(m− 2) .
Let (Xt)t∈Z a real-valued stationary process. We use a quan-tile transform to assume that the marginal distribution of this se-quence is the uniform law on [0, 1] . Here (B(x))x∈[0,1] is the depen-dent analogue of a Brownian bridge, B is centered, Gaussian process:EB(x)B(y) =
P∞k=−∞ (P (X0 ≤ x, Xk ≤ y)− P (X0 ≤ x)P (Xk ≤ y)) .
Theorem 2 Suppose that the stationary sequence (Xt)t∈Z has uniform marginaldistribution and is either η -weakly dependent with ηr = O(r−15/2−ν) , or κ -weaklydependent κr = O(r−5−ν) , for some ν > 0 . Then the following empirical func-tional convergence holds true in the Skohorod space of real-valued cadlag functionson the unit interval, D([0, 1]) :
√n (Fn(x)− F (x))) → B(x) .
[1] Dedecker, J., Doukhan, P. (2003) A new covariance inequality and applications,Stoch. Proc. Appl. Vol. 106, N 1, 63-80.
[2] Doukhan, P., Louhichi S. (1999) A new weak dependence condition and applicationsto moment inequalities. Stoch. Proc. Appl. Vol. 84, 313-342.
[3] Doukhan, P. Teyssiere, G., Winant, P. (2005) Vector valued ARCH infinity processes;submitted.
[4] Doukhan, P., Wintenberger, O. (2005) Invariance principle for new weakly dependentstationary models under sharp moment assumptions. Submitted for publication.
[5] Rosenblatt, M. (1985) Stationary processes and random fields. Birkhauser, Boston.
Laboratoire de Statistique du CREST and SAMOS, Statistique Appliquee etModelisation Stochastique, Universite Paris 1, FranceMailing Address: Laboratoire de Statistique du CREST, Timbre J3403, avenuePierre Larousse, 92240 MALAKOFF, France. E-mail: [email protected]
126
Optimal control of company assets on theassumption of sales stabilization.
Elejko T.Y., Nakonechny A.N., UkraineEvery firm from the moment of it’s foundation goes through a few stages of
development and for every successful company comes the period when it almostcompletely occupies it’s market share. It means that average sales of this firmstabilized on some level. In such case there’s no sense for the firm to increase it’sproduction capacity because of new facility wouldn’t be engaged. So the managersshould decide what to do with the money that were used for firm developing before.In other words it should be founded the optimal control of revenue distribution,that will maximize cost of firm’s share capital when average sales stabilize.
We’ll look over this problem in terms of marcovian models of firm developing(MMFD)[1]. According to MMFD state of the company in each period xi isprecisely defined by the value of company’s own asset Ti and revenue during thisperiod Ei . Thus xi = 〈Ti, Ei〉 .
Function ϕi : Xi → [0; 1] is called the simple strategy in period i wereXi = ∪xi . Such simple strategy defines the relative part of the revenue thatshould be spent on credit repayment during corresponding period if in this periodcompany is in state xi . Let π denote the policy. The use of policy π means mak-ing sequential decisions, using corresponding simple strategy ϕi , i = 1 . . . ,∞ :π = ϕ1ϕ2ϕ3 . . . .
We should note that under total restriction of sales our MMFD is ho-mogeneous. It means that for our model exists the stationary optimal policyπopt = ϕ′ϕ′ϕ′ . . . . So in order to maximize the firm’s share capital when aver-age sales stabilize we should use the same simple strategy ϕ′ in every period.
In practice it’s very difficult to solve the fundamental equation for finding theoptimal stationary strategy, therefore we developed the method of constructionso called ε -optimal policy and showed that this policy also could be stationary:πεopt = ϕ′εϕ
′εϕ′ε . . . . So if we’ll use our ε -optimal policy πεopt the average cost of
firm’s share capital will be less than maximal possible one not more than ε . Insuch a way we built the approximation of optimal policy that is very simple inpractical use. In article [2] was considered the MMFD under total sales restrictionand solved the problem of stochastic optimization, in other words there were foundthe constant percent of firm’s revenue that should be spent for credit repaymentin every period, for maximizing cost of firm’s share capital. Such policy is usuallycalled the simplest optimal policy.
[1] Nakonechny A.N., Elejko T.Y., Research of firm’s share capital cost in dependenceof capital structure// Papers of NAS of Ukraine - 2005. - 1. - с.65-70.
[2] Elejko T.Y., Optimal policy of capital structure changing under total sales restric-tion// Papers of NAS of Ukraine. - 2005. - 4. - с.59-64.
[3] Dynkin E.B., Uskevich A.A., Control markov processes and their applications. -Moskov:Nauka, 1975.
[4] Ross S.A., Westerfiled R.W., Jordan B.D. Fundamentals of Corporate Finance(Fifth Edition). - Toronto: McGraw-Hill, 2001.
Institute of Cybernetics; e-mail: [email protected]
127
Two-component contact model
Denis Filonenko, Ukraine
We consider an example of the two-component contact model with interactionand discover asymptotic behaviour of the first and second correlation functions fornon-equilibrium dynamics. Our system has two types of of particles in Rd (d ≥ 3) :(+) and (−) . (−) -system is the independent one-component contact model. (+) -system is the sum of the independent contact model and the interaction term whichdescribes the birth of (+) -particles by linear rate influence of (−) -particles.
The formal generator of the dynamics has the following form:
(LF )`γ+, γ−
´=
Xx∈γ+
ˆF`γ+ \ x, γ−
´− F
`γ+, γ−
´˜
+λ1
ZRd
0@ Xx′∈γ+
a1
`x− x′
´1AˆF `γ+ ∪ x, γ−´− F
`γ+, γ−
´˜dx
+Xy∈γ−
ˆF`γ+, γ− \ y
´− F
`γ+, γ−
´˜
+λ2
ZRd
0@ Xy′∈γ−
a2
`y − y′
´1AˆF `γ+, γ− ∪ y´− F
`γ+, γ−
´˜dy
+λ
ZRd
0@Xy∈γ−
a (x− y)
1AˆF `γ+ ∪ x, γ−´− F
`γ+, γ−
´˜dx
where a1, a2, a are even non-negative integrable functions on Rd; λ, λ1, λ2 > 0 .We found asymptotic of the first correlation functions k±t of (+) and (−) -
systems and of the second correlation functions k++t ((+) -system) , k−−t ((−) -
system) , k+−t (characterisation of interaction).
Kyiv-Mohyla Academy, Ukraine,Department of mathematical physics,Institute of Mathematics NAS of Ukraine,3, Tereshchenkivska Str. 01601 Kyiv-4, Ukrainee-mail: [email protected]
128
Equilibrium Glauber dynamics of continuousparticle systems as a scaling limit of Kawasaki
dynamicsDmitri Finkelshtein, Ukraine; Yuri Kondratiev, Germany; Eugene Lytvynov, U.K.
A Kawasaki dynamics in continuum is a dynamics of an infinite system ofinteracting particles in Rd which randomly hop over the space. In this paper, wedeal with an equilibrium Kawasaki dynamics which has a Gibbs measure µ asinvariant measure. About µ we assume that it corresponds to an activity para-meter z > 0 and a potential of pair interaction φ . The generator of the Kawasakidynamics is given, on an appropriate set of cylinder functions, by
(HF )(γ) = −Xx∈γ
ZRd
dy a(x− y) exp
24− Xu∈γ\x
φ(u− y)
35× (F (γ \ x ∪ y)− F (γ)), γ ∈ Γ. (1)
Here, Γ denotes the configuration space over Rd , i.e., the space of all locally finitesubsets of Rd , and, for simplicity of notations, we just write x instead of x .About the function a(·) in (1) we assume that it is non-negative, integrable andsymmetric with respect to the origin. The factor a(x−y) exp
h−Pu∈γ\x φ(u− y)
iin (1) describes the rate with which, given a configuration γ ∈ Γ , a particle x ∈ γjumps to y .
Under very mild assumptions on the Gibbs measure µ , it was proved in [3]that there indeed exists a Markov process on Γ with cadlag paths whose generatoris given by (1). We assume that the initial distribution of this dynamics is µ , andperform the following scaling of this dynamics. For each ε > 0 , we consider theequilibrium Kawasaki dynamics whose generator is given by formula (1) in whicha(·) is replace by the function aε(·) := εda(ε·). We denote this generator by Hε ,and study the limit of the corresponding dynamics as ε→ 0 (Kac-type limit).
Informally, we expect that, in the limit, only jumps of infinite length willsurvive, i.e., jumps from a point to ‘infinity’ and from ‘infinity’ to a point. Thus,we expect to arrive at a Glauber dynamics in continuum, i.e., a birth-and-deathprocess in Rd , cf. [1, 3]. In fact, heuristic calculations show that the limitingGlauber dynamics has the generator
(H0F )(γ) = −αXx∈γ
(F (γ \ x)− F (γ))
− α
ZRd
z dx exp
"−Xu∈γ
φ(u− x)
#(F (γ ∪ x)− F (γ)), (2)
where α = z−1k(1)µ
RXa(x) dx, k
(1)µ being the first correlation function of the
measure µ . Thus, α describes the rate with which a particle x ∈ γ dies, whereas
129
αz exph−Pu∈γ φ(u− x)
idescribes the rate with which, given a configuration γ ,
a new particle is born at x ∈ Rd \ γ . The existence of a Markov process on Γwith cadlag paths, whose generator is given by (2), was proved in [1] (see also [3]).
The main results of this paper are as follows:• For any stable potential φ in the low activity-high temperature regime, the
generators Hε converge to the generator H0 . The convergence is on the setof exponential functions, in the L2(Γ, µ) -norm.
• For any positive potential φ in the low activity-high temperature regime,the finite-dimensional distributions of the Kawasaki dynamics with genera-tor Hε and initial distribution µ weakly converge to the finite-dimensionaldistributions of the Glauber dynamics with generator H0 and initial distri-bution µ .
To prove the first main result, we essentially use the Ruelle bound on thecorrelation functions of the measure µ , as well as the integrability of the Ursell(cluster) functions of µ , proved by Brox [1]. To derive from here the convergenceof the finite-dimensional distributions of the dynamics, we additionally need thatthe set of finite sums of exponential functions forms a core for the generator ofthe limiting dynamics, H0 . For this, we use a result from [1] on a core for H0 ,which holds under the assumptions of positivity of the potential φ .
References
[1] T. Brox, “Gibbsgleichgewichtsfluktuationen fur einige Potentiallimiten”, PhDthesis, Universitat Heidelberg, 1980.
[2] Yu. G. Kondratiev and E. Lytvynov, Glauber dynamics of continuous particlesystems, Ann. Inst. H. Poincare Probab. Statist. 41 (2005), 685–702.
[3] Yu. G. Kondratiev, E. Lytvynov, and M. Rockner, Equilibrium Glauber andKawasaki dynamics of continuous particle systems, Preprint, 2005, availableat www.arxiv.org/math.PR/0503042
[Dmitri L. Finkelshtein]Institute of Mathematics, National Academy of Sciences of Ukraine,3 Tereshchenkivska Str., Kiev 01601, Ukrainee-mail: [email protected][Yuri G. Kondratiev]Fakultat fur Mathematik, Universitat Bielefeld,Postfach 10 01 31, D-33501 Bielefeld, Germany;BiBoS, Univ. Bielefeld, Germany;Kiev-Mohyla Academy, Kiev, Ukrainee-mail: [email protected][Eugene W. Lytvynov]Department of Mathematics, University of Wales Swansea,Singleton Park, Swansea SA2 8PP, U.K.e-mail: [email protected]
130
Stationary solutions of a differenceequation in a Banach space
Gorodnii Mykhaylo, Ukraine
Let (X, ‖ ·‖) be a complex Banach space, L(X) the Banach space of boundedlinear operators on X with the operator norm, A a closed operator on X , andAn : n ∈ Z a sequence of operators of the space L(X) with
Pn∈Z ‖An‖ <∞ .
Definition. A sequence of X -valued random elements η := ηn : n ∈ Zdefined on a complete probability space (Ω, F, P ) is said to be stationary if
1) supn∈Z
E‖ηn‖2X <∞;
2)∀n ∈ Z : Eηn = Eη0;
3)∀n, k ∈ Z ∀x∗, y∗ ∈ X∗ :
cov(< ηn+k, x∗ >,< ηk, y
∗ >) = cov(< ηn, x∗ >,< η0, y
∗ >).
Here X∗ is the dual space of X .By stationary solution of the difference equation
(1) Aξn =Xk∈Z
Akξn+k + ηn, n ∈ Z,
corresponding to the stationary sequence η , we mean a stationary sequence ξthat satisfies (1) with probability 1.
Theorem. The difference equation (1) has a unique stationary solution ξ forany stationary sequence η if for any z ∈ C, |z| = 1, the operator A−
Pn∈Z Anz
n
has a continuous inverse.
Kyiv T. Shevchenko University,Dept. of Mathematics,Volodymyrska, 64, 01601 Kyiv, Ukrainee-mail: [email protected]
131
On the behaviour of risk processafter the ruin time
Dmytro Gusak, Ukraine
Last time the interest to different characteristics of the risk process definingits behaviour after the ruin time is constantly growing. All these characteristicsare connected with the overjumps functionals, which were studied in [1] - [4].
To define mentioned characteristics, some of which are introduced in [5, 6] weconsider only the Claim surplus process with initial capital u > 0 ,
ζ(t) = S(t)− ct, S(t) =Xk≤ν(t)
ξk, P ξk > 0 = 1,
ν(t) - the simple Poisson process with the rate λ > 0 .We denote the functionals of ζ(t) :
τ+(u) = inf t > 0 : ζ(t) > 0 - the ruin time;γ+(u) = ζ(τ+(u))− u - the severity of ruin;γ+(u) = u− ζ(τ+(u)− 0) - the surplus prior to ruin;γ+u = γ+(u) + γ+(u) - the claim which causes the ruin;ζ+(t) = sup
0≤s≤tζ(t) - the extrema of ζ(t) ;
Z+(u) = supτ+(u)≤t<∞
ζ(t) - the total maximal deficit;
τ ′(u) = inf˘t : t > τ+(u), ζ(t) < u
¯the first return time of ζ(t) under u > 0 ;
T ′(u) =
τ ′(u)− τ+(u), τ+(u) <∞,0, τ+(u) = ∞.
T ′(u) is called in [5] “the red time” and it defines the duration of stay of ζ(t)above the critical level u > 0 .
The relations for distributions of all mentioned functionals are established onthe basis of results obtained in [2, 4] for the joint moment generating function(m.g.f.) of the four functionals˘
τ+(u), γ+(u), γ+(u), γ+u
¯.
In terms of the last m.g.f. we obtain also the relation for the density of thegeneral multivariate ruin function
ϕs(u, dx, dy) = Ehe−sτ
+(u), γ+(u) ∈ dx, γ+(u) ∈ dy, τ+(u) <∞i,
and for the limit density (when m = E ζ(1) < 0)
ϕ0(u, dx, dy) = P˘γ+(u) ∈ dx, γ+(u) ∈ dy, τ+(u) <∞
¯.
132
Let θs be the exponentially distributed random variable with the parameters > 0 , then
ϕs(u, dx, dy) = Pζ+(θs) > u, γ+(u) ∈ dx, γ+(u) ∈ dy,
ϕ0(u, dx, dy) = Pζ+ > u, γ+(u) ∈ dx, γ+(u) ∈ dy,where ζ+ = sup
0≤t≤∞ζ(t) (the absolute maximum of ζ(t)) has nondegenerate dis-
tribution, if m < 0 .
References
[1] D.V. Gusak, V.S. Korolyuk, On the first passage time across a given level forprocess with independent increments,Theory of Prob. and Appl. 13 (1968),no.3, pp. 448-456(transl. from Russian).
[2] D.V. Gusak, On the joint distribution of the first exit time and exit value forhomogeneous process with independent increments, Theory of Prob. and Appl.14 (1969), no.1, pp. 14-23(transl. from Russian).
[3] D.V. Gusak, V.S. Korolyuk, On the joint distribution of the process with sta-tionary increments and its maximum, Theory of Prob. and Appl. 14 (1969),no.3, pp. 400-409(transl. from Russian).
[4] D.V. Gusak, The distributions of overjumps functionals of the homogeneousprocess with independent increments, Ukr.Math.Journal, 54 (2002), no. 3, pp.371-392 (transl. from Ukrainian).
[5] T. Rolsky, H. Shmidly, V. Shmidt, J.Teugels(1999) Stochastic Processes forInsurance and Finance, John Wiley, New York, pp.654.
[6] S. Asmussen(2000) Ruin Probabilities, Word Scientist, Singapore, pp.385.
Institute of Mathematics Ukr. Nat. Ac. Sci.,3 Tereshenkivska str., 01601 Kiev, Ukrainee-mail: [email protected]
133
On robust utility maximizationAlexander Gushchin, Russia
1. Robust utility functionals are numerical representations of preferences ofthe investor who is averse against both risk and uncertainty about the underlyingprobabilistic model. We consider a general arbitrage pricing model, i.e. a proba-bility space (Ω,F ,P) and a set A of random variables interpreted as the set ofpossible incomes. It is assumed that A is a convex cone in L0(P) and every ξ ∈ Ais bounded from below. The utility function U : R → R∪−∞ is assumed to be anupper semicontinuous increasing concave function; r := − inf x : U(x) > −∞ .The set Q of subjective measures is convex, P(B) = 0 iff Q(B) = 0 for everyQ ∈ Q , and the set dQ/dP : Q ∈ Q is closed with respect to convergence inP -probability. The robust utility maximization problem is to
maximize infQ∈Q
EQU(x+ ξ) over ξ ∈ A, (1)
where x ∈ R is the initial endowment (here and below expectation is assumed tobe −∞ if it is not well defined). The value function of the problem is
u(x) = supξ∈A
infQ∈Q
EQU(x+ ξ).
The function u : R → R ∪ −∞ ∪ +∞ is concave and increasing, u ≥ U . Ourpurpose is to obtain a representation for u by describing its convex conjugatev(y) = supx∈R[u(x)− xy] .
2. Let V (y) = supx∈R[U(x)−xy] , y ∈ R , then V : R → R∪+∞ is a properlower semicontinuous convex function with domV ⊆ R+ . Put also
V(s, t) = supx,y∈R : y≤U(x)
(ty − sx), sothat V (s) = V(s, 1).
Denote by ba the dual space of the space L∞ of bounded measurable func-tions, i.e. ba is the space of bounded finitely additive set functions µ : F → R ; thecountably additive and the purely finitely additive components of µ ∈ ba are de-noted by µc and µa respectively. Define the V -divergence JV (µ, ν) of µ , ν ∈ baas follows:
JV (µ, ν) = supξ,η∈L∞ : η≤U(ξ)
[ν(η)− µ(ξ)].
Theorem 1 (a) The function JV (µ, ν) is proper, convex, and lower semicontin-uous in the topology σ(ba× ba,L∞ × L∞) .
(b) JV (µ, ν) = supPni=1 V(µ(Bi), ν(Bi)), where the supremum is taken over
all finite partitions of Ω into disjoint sets B1, . . . , Bn ∈ F .(c) If µ and ν are (positive countably additive) finite measures dominated by
a measure ρ then
JV (µ, ν) =
ZV(dµ/dρ, dν/dρ) dρ.
(d) JV (µ, ν) = JV (µc, νc) + JV (µa, νa).
134
The property (c) says that our definition coincides with the standard definitionof the V -divergence of finite measures if V takes finite values on (0,+∞) .
3. We return to the problem (1). Put C =`A − L0
+(P)´∩ L∞(P) . Introduce
the set R of “separating” finitely additive set functions by
R = µ ∈ ba+(P) : µ(ξ) ≤ 0 ∀ξ ∈ C,
where ba+(P) = µ ∈ ba : µ(B) ≥ 0 ∀B ∈ F and µ(B) = 0 if P(B) = 0 . Weexclude the case R = 0 (in other words, C = L∞(P)) from the considerationsince it can be treated in a trivial way. Then Ry = µ ∈ R : µ(Ω) = y 6= ∅ forevery y > 0 , and we can define
v(y) = infµ∈Ry,Q∈Q
JV (µ,Q), if y ≥ 0, (2)
and v(y) = +∞ otherwise. The function v is convex and lower semicontinuousand v ≥ V . Note that the infimum in (2) is attained since both Ry and Q arecompacts in the topology σ(ba(P), L∞(P)) .
Theorem 2 (a) If v(y) 6≡ +∞, then
u(x) = miny∈R
[v(y) + xy] = minµ∈R,Q∈Q
[JV (µ,Q) + xµ(Ω)], x > −r.
(b) If v(y) ≡ +∞, then u(x) = +∞ for x > −r .
If r = ∞ , then JV (µ,Q) <∞ only if µ is countably additive and absolutelycontinuous with respect to Q , see Theorem 1. Thus, the representation (2) is notnew, cf. Schachermayer [3] for the case Q = P and Follmer and Gundel [1].However, the proof of the existence of the “least favorable pair” of measures isquite elementary in our approach. If r <∞ , then the representation (2) for v isnew and has several advantages, cf. Kramkov and Schachermayer [2] for the caseQ = P and Schied and Wu [4]. Note also that the assumptions on the marketand the utility function are more restrictive in all the papers mentioned above.
References
1. Follmer, H. and Gundel, A. Robust projections in the class of martingalemeasures. To appear in Illinois Journal of Mathematics, 2006.
2. Kramkov, D. and Schachermayer, W. The asymptotic elasticity of utilityfunctions and optimal investment in incomplete markets. Ann. Appl.Probab., 1999, v. 9, no. 3, p. 904–950.
3. Schachermayer, W. Optimal investment in incomplete markets whenwealth may become negative. Ann. Appl. Probab., 2001, v. 11, no. 3,p. 694–734.
4. Schied A., Wu C.-T. Duality theory for optimal investments under modeluncertainty. Statistics & Decisions, 2005, v. 23, no. 3, p. 199-217.
Steklov Mathemtical Institute,Gubkina 8, 119991 Moscow, Russiae-mail: [email protected]
135
On generalized n-th derivative summable insquare functions in the Hilbert space with
Gaussian measureHajiyev V.H., Azerbaijan
Let X be a Hilbert space with the scalar product (x, y) ,x, y ∈ X,=− σ be analgebra of Borel sets from X, µ be a Gaussian measure on = with thecharacteristic functional ϕ0(z) = exp −1/2 (Bz, z) , where B is a correlationoperator. Denote by L2 (X,µ) a spae of sumable in square functions in X . TheFourier transformation of the function f(x) ∈ L2 (X,µ) is defined by the formϕ(z) =
Rei(z,x)f(x)µdx . Let X be complex expansion of X whose elements are
formal sums x+ iy, x, y ∈ X with real linear operations over them and scalarproduct defined by the formula(x1 + iy1, x2 + iy2)k = [(x1, x2) + (y1, y2)] + i[(y1, y2)− (x1, y2)] , then X will bea complex Hilbert space, and X will be its subspace. The functionseϕ0 (x1 + y1) =
Rei(x1+λy1,x)kµ(dx) and eϕ0 (x1 + y1) =
Rei(x1+λy1,x)kf(x)µ(dx)
are the entire analytical functions with respect to complex variable λ andcoinciding with ϕ0(z) and ϕ(z) at real numbers λ . So the functions ϕ0(z) andϕ(z) are extended on X and ϕ0 (x+ λy) , ϕ (x+ λy) will be the entireanalytical functions with respect to λ at any fixed x, y ∈ X .Let’s introduce the following differential operators
Pn`ddz
´ϕ(z) =
nPk=1
nPi1...in=0
ci1...ikϕ(k)(z; ei1 ...eik )
corresponding to the polynomial function
Pn(x) =nPk=1
nPi1...in=0
ci1...ik (x, ei1)...(x, eik )
where n ≥ 1, Ci1...ik are numbers,x, ei1 , ei2 , ..., eik ∈ X are arbitrary.
Qb1...bl`ddz
´ϕ(z) =
lQi=1
ϕ(i)(z; bi) , and P(k)n,ai1 ...aik
(x) are k -th derivatives of
Pn(x) in the direction of ai1 , ...aik .In this abstract we show the necessary and sufficient condition on the smoothfunction ϕ(z) , z ∈ X such that the function f(x) ∈ L2 (X,µ) has the derivativef (n) (x; a1, a2, ...an) , where a1, a2, ...an ∈ BX , defined almost everywhere andf (n) (x; a1, a2, ...an) ∈ L2 (X,µ) .Theorem. In order to f(x) ∈ L2 (X,µ) has the derivativef (n) (x; a1, a2, ...an) ∈ L2 (X,µ) , where a1, a2, ...an ∈ BX , it’s necessary andsufficient that
|lϕ,a1,...an(Pn)|2 ≤ Ca1,...an
ZP 2n(x)µ(dx),
where lϕ,a1,...,an(Pn) =
=hP
ikP(k)n,ai1 ,...,aik
`1idz´QB−1aj1 ,...,B
−1ajn−k
`ddz
´iϕ(z)ϕ(z)′z=0
where the summable is led on all choice of (ai1 , aik ) such that(ai1 , aik ) ^
`aj1 , ajn−k
´= (a1, a2, ...an) and Ca1,a2,...,an is constant.
Baku State University
136
Modelling of Galton-Watson branchingprocesses with random errors
Hasratova M.H., Azerbaijan
Galton -Watson branching processes is considered, in which at the appearancetime of new populations occurs the random errors ξi , i = 1, 2, 3... . The randomerrors can take both positive, and negative values. It is assumed that therandom errors at each populations generate. The sequence of independentequally distributed random values with Eξi and Eξ2i = σ2
i where - σ2i are
unknown differences, σ2i .
It is suggested the approach for modeling of such processes. As investigatedcharacteristic the average number of particles in system is chosen. The numericalresults of investigated characteristic are obtained, and the new universal methodof construction of confidential band is also proposed for indicated characteristicof confidential band. Programs in C++ are developed for modelling of suchprocesses, as well as for construction of confidential band.
Baku State Universitye-mail: [email protected]
137
Multidimensional model of the diffusion withmembrane at the hyperplane which has aproperty of partial reflection and delay
Serhiy Huran, Ukraine
Let Di = x : x = (xl, . . . , xn) ∈ Rn, (−1)ixn > 0 , i = 1, 2 , be domains inthe finite-dimensional Euclidean space Rn , n > 2 , S = x : x ∈ Rn, xn = 0 bethe common boundary of Di . Assume that in each of the domains Di , i = 1, 2 ,the diffusion process with the forming operator
Li =1
2
nXk,j=1
b(i)kj
∂2
∂xk∂xj, i = 1, 2, (1)
is defined, where Bi =“b(i)kj
”nk,j=1
, i = 1, 2 , are constant symmetric and positive
matrices.We consider the problem of existence of the semigroup of operators Tt , t > 0 ,
that describes the continuous Feller process in Rn such that in D1 and D2 itcoincides with diffusion processes formed by L1 and L2 , and at the S -surface thefunction Ttϕ(x) in addition the Fellers condition satisfy the following adjunctioncondition:
σ · ∂Ttϕ(x)
∂t= q1
∂Ttϕ(x−)
∂N1+ q2
∂Ttϕ(x+)
∂N2, t > 0, x ∈ S, (2)
where σ, ql, q2 are real numbers such that σ > 0 , q1 6 0 , q2 > 0 , q2 − q1 > 0 ,Ni = Biν , i = 1, 2 , ν = (0, . . . , 0, 1) ∈ Rn is the co-normal vectors to S at the
point x . And by∂Ttϕ(x−)
∂N1,∂Ttϕ(x+)
∂N2we denote the non-tangent limits of the
functions∂Ttϕ(z)
∂N1and
∂Ttϕ(z)
∂N2, when z is approaching x ∈ S from the sides of
D1 or D2 respectively.The integral representation of the required semigroup is obtained by the meth-
ods of classical theory of potential as a solution of the respective conjunction prob-lem for the second order linear parabolic equation with discontinuous coefficients.
Moreover, we prove that the generated by the semigroup of operators Markovprocess could be considered as a diffusion process in the sense of Kolmogorov(see [1]).
References
[1] M. I. Portenko, Diffusion processes in the domain with membranes. Instituteof Mathematics of National Academy of Sciences of Ukraine, Kiyv, 1995.
Department of Mathematics,Ivan Franko Lviv National University,Universytetska 1, Lviv, 79000, Ukrainee-mail: [email protected]
138
Risk of forecasting of autoregressive timeseries with missing data
Aliaksandr Huryn, Belarus
Let the observed time series be described by the AR(1) model of autoregressionof the first order [1]:
yt+1 = byt + ut+1, t ∈ Z, (1)
where b ∈ (−1, 1) is an unknown coefficient, ut : t ∈ Z are i.i.d. Gaussianrandom variables with zero mean: Lut = N
`0, σ2
´, 0 < σ2 < +∞ . The
time series yt contains missing values. For each time moment t ∈ Z the non-random binary variable (called “missing pattern”) Ot ∈ 0, 1 is given, whereOt = 1 , if yt is observed; 0, if yt is a missing value . Denote the minimal and themaximal time moments in which the process is observed: t− = min t : Ot = 1 ,t+ = max t : Ot = 1 ; without loss of generality t− = 1 , t+ = T .
It is well known that if the model parameter b is known then the best onestep forecast (in the sense of minimum mean square error) is yT+1 = byT [1],and its risk is rT = E
˘(yT+1 − yT+1)2
¯= σ2 . Let the parameter b be unknown.
Denote Gk = Covyt+k, yt = bkσ2
1−b2 , k ∈ 0, 1 , construct the estimator of model
parameter b [2]: b = G1G0, Gk =
PT−kt=1 yt+kytOt+kOtPT−k
t=1 Ot+kOt, and the “plug-in” forecast:
yT+1,“plug-in” = byT .
Theorem 1 Let the model (1) take place, the asymptotic as-sumption on “missing pattern” be satisfied at T → ∞ :
1T−|s|
PT−1t,t′=1Ot+kOtOt′+k′Ot′δt−t′,s → ϑs,k,k′ ∈ [0, 1], s ∈ Z, k, k′ ∈ 0, 1 , and
ϑ0,0,0, ϑ1,0,0 6= 0 , where δi,j is a Kroneker symbol. Then the following asymptoticexpansion of the mean square risk of the “plug-in” forecast takes place:
rT,“plug-in” = E˘
(yT+1,“plug-in” − yT+1)2¯
= σ2 +1
T
2b2G0
ϑ20,0,0
∞Xs=−∞
b2|s|ϑs,0,0−
4bG0
ϑ0,0,0ϑ1,0,0
∞Xs=−∞
b|s−1|+|s|ϑs,0,1 +G0
ϑ21,0,0
∞Xs=−∞
(b|s−1|+|s|)ϑs,1,1
!+ o
„1
T
«.
Corollary 1 Let the model (1) take place and there are no missing values in thedata, i.e. Ot = 1 , t ∈ 1, . . . , T . Then the following asymptotic expansion of themean square risk takes place: rT,“plug-in” = σ2 + σ2
T+ o
`1T
´.
[1] T. Anderson, The Statistical Analysis of Time Series, Wiley, New York (1971).
[2] Yu. Kharin, A. Huryn, Austrian Journal of Statistics, 34, 2, 163–174 (2005).
Belarusian State University,220030, Belarus, Minsk, Nesavisimosti ave., 4e-mail: [email protected]
139
On a martingale related to the branchingrandom walk
Aleksander Iksanov, Ukraine
Let M be a point process on R . The variable L := M(R) may be deter-ministic or random, finite or infinite with positive probability. By the branchingrandom walk is meant the sequence of point processes Mn, n = 0, 1, . . . which aredefined as follows: for any Borel set B ⊂ R
M0(B) = 10∈B, Mn+1(B) :=Xr
Mn,r(B −An,r), n = 0, 1, . . . .
Here An,r are the points of Mn , and Mn,r are independent copies of theM . We only consider the supercritical BRW. Therefore, if PL < ∞ = 1 it isadditionally assumed that EL > 1 .
Assume that for some γ > 0
m(γ) := EZ
ReγxM(dx) ∈ (0,∞)
and setWn := m(γ)−n
ZReγxMn(dx).
The sequence (Wn, σ(M1, . . . ,Mn)), n = 1, 2, . . . is a nonnegative martingaleand therefore it converges a.s. The limit random variable W , say, is positive withpositive probability iff Wn is uniformly integrable. Our first result provides acriterion of uniform integrability.
Theorem 1 [1] The martingale Wn is uniformly integrable if and only if
limn→∞
Z1Z2 . . . Zn = 0 a.s.;
Z(1,∞)
x log xR x0
P− logZ > ydydPW1 ≤ x <∞,
where Z1, Z2, . . . are independent copies of a random variable Z whose distribu-tion is defined by the equality
Ek(Z) = EZ
R
eγx
m(γ)k(
eγx
m(γ))M(dx),
which is assumed to hold for any nonnegative bounded Borel function k(x) .
As soon as the conditions for uniform integrability of Wn are known it isnatural to ask about the rate of a.s. convergence of Wn to W (under these condi-tions). Let a : R+ → R+ be a function that regularly varies at ∞ with exponent
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α > −1 . If α = 0 , we additionally assume that a is eventually nondecreasing.The next result points out sufficient conditions for a.s. convergence of the series
∞Xn=0
a(n)(W −Wn). (1)
Clearly, this should be regarded as a rate of convergence result.
Theorem 2 [2] Assume that
E logZ ∈ (−∞, 0) and EW1 log+W1 <∞, (2)
and the distribution of logZ is non-arithmetic. The conditions
E(log+ Z)3a(log+ Z) <∞ and EW1(log+W1)2a(log+W1) <∞ (3)
are sufficient for a.s. convergence of (1).
We think that the first inequality in (3) can be weakened toE(log+ Z)2a(log+ Z) < ∞ . If the conjecture is correct then it can be checkedthat the following equivalence should be true˛
˛∞Xn=0
a(n)(W −Wn)
˛˛ <∞ a.s.⇔ EW log+Wa(log+W ) <∞.
It can be verified that the conjecture holds true for two particular cases: if Wn
reduces to the normalized Galton-Watson process or if the positions of childrenof each individual are displaced to the left from their parent’s position.
The last result deals with existence of some specific moments of the W .
Theorem 3 [2] If (2) holds, then
EW log+Wa(log+W ) <∞ iff EW1(log+W1)2a(log+W1) <∞.
Other moment results were obtained in [3], and these will be discussed too.
References
[1] Iksanov, A. M. (2004). Elementary fixed points of the BRW smoothingtransforms with infinite number of summands. Stoch.Proc.Appl. 114, 27-50.
[2] Iksanov, A. M. (2006). On the rate of convergence of a regular martingalerelated to the branching random walk, (in Ukrainian). Ukrainian Math. J.58(3), 326-342.
[3] Iksanov, A. M. and Rosler, U. (2006). Some moment results about thelimit of a martingale related to the supercritical branching random walk andperpetuities. Ukrainian Math. J. 58(4).
Faculty of Cybernetics,National T.Shevchenko University, Kiev-01033, Ukrainee-mail: [email protected]
141
On asymptotical extinction of linearnonhomogeneous stochastic differential
equation systems
Ilchenko A. V., Ukraine
Consider the following system of stochastic differential equations
dx(t) =`A0x(t) + f0(t)
´dt+
mXk=1
`Akx(t) + fk(t)
´dwk(t), (1)
where Ak are real n×n matrices; fk(t) =`fk1(t), . . . , fkn(t)
´, t ≥ 0 are columns
of vector-valued functions; wr(t) , t ≥ 0 are standard scalar independent Wienerprocesses. The Euclidean norm of vectors from Rn is designated as ‖x‖ .
Denote by Hts the operator-valued solution for the homogeneous system
dHts = A0H
tsdt+
mXk=1
AkHtsdwk(t), Hs
s = I, s ≤ t.
Let for some p0 > 0 there exist D = D(p) > 0 and λ = λ(p) > 0 such that
sup‖x‖=1
E‚‚Ht
sx‚‚p ≤ De−λp (t−s), p ∈ (0, p0). (2)
Theorem 1 Suppose that the condition (2) is fulfilled and fk(t) are continuousfunctions such that limt→+∞ fk(t) = 0. Then solutions of the system (1) vanishin probability.
Theorem 2 Suppose that the condition (2) is fulfilled and fk(t) are continuousfunctions such that
∞Xj=0
jXi=0
e−γ(j−i)p supi≤u≤i+1
‚‚fk(u)‚‚p < +∞, k = 0,m
for some p, γ : 0 < p < p0 , 0 < γ < λ . Then solutions of the system (1) vanishwith probability 1.
Another kind of behavior of the system (1) was considered in [1,2].
[1] A. V. Ilchenko, Stochastically bounded solutions of the linear nonhomo-geneous stochastic differential equation, Theor. Probability and Math.Statyst., No. 68(2003), p. 48-55.
[2] A. V. Ilchenko, Stochastically bounded solutions of the linear nonho-mogeneous stochastic differential equation system, Theory Of StochasticProcesses, vol. 9(25) (2003), no. 1-2, p. 65-72.
Kyiv National T. Shevchenko University, Ukrainee-mail: [email protected]
142
The Levy distance as a measure of convergenceK.-H. Indlekofer, Germany; O. I. Klesov, Ukraine
We study the deviation between two distribution functions and estimate it viathe Levy distance between the functions. Analogous estimates are well known forthe case where the uniform distance is used instead of the Levy distance. Howeverthe latter is more natural in various cases.
For example let a sequence of distribution functions Fn, n ≥ 1 weakly con-verge to the limit law F . Then it is known that the Levy distance Ln = L(Fn, F )also approaches zero. On the other hand if there are discontinuities of the limitlaw F , then the uniform distance δn = δ(Fn, F ) not necessarily tends to 0 inthis case. In such a case the nonuniform estimates like |Fn(x)− F (x)| ≤ d(δn, x)where d is a certain function of two arguments are usually meaningless, sincethe right hand side, as a rule, does not approach zero even x is a point of con-tinuity of the limit despite its left hand side does! The above function d usuallyis such that limt↓0 d(t, x) = 0 and estimates where the Levy distance substi-tutes the uniform distance would be helpful, since both sides of the inequality|Fn(x)− F (x)| ≤ d(Ln, x) then approach zero.
Just to have a feeling of what kind of functions d are used for the uniformmetric, consider a result of Esseen: let Xn, n ≥ 1 be a sequence of independentrandom variables such that
EXn = 0, EX2n = σ2
n <∞, n ≥ 1.
Put
s2n =
nXk=1
σ2k, Fn(x) = P
„X1 + · · ·+Xn
sn< x
«.
If δn ≤ 12
for n > n0 , then there exists an absolute constant C such that
|Fn(x)− Φ(x)| ≤ min
(δn;C
δn ln 1δn
1 + x2
).
Here Φ stands for the standard Gaussian distribution function.In particular, we prove an analog of this result for the Levy distance instead
of the uniform metric by extending the Kolodyazhniı method to the case of theLevy distance.
Fachbereich Mathematik und Informatik, Universitat Paderborn,Warburger Straße 100, D–33098 Paderborn,Bundesrepublik Deutschlande-mail: [email protected] of Mathematical Analysis and Probability Theory,National Technical University of Ukraine (KPI),pr. Peremogy, 37, Kyiv 03056, Ukrainee-mail: [email protected]
143
Financial market models with memory
Akihiko Inoue, Japan
We consider a financial market model M driven by an Rn -valued Gaussianprocess with stationary increments which is different from Brownian motion. Thisdriving noise process Y (t) = (Y1(t), . . . , Yn(t))′ consists of n independent compo-nents, and the j th component Yj(t) is defined by the autoregressive type equation
dYj(t)
dt= −
Z t
−∞pje
−qj(t−s) dYj(s)
dsds+
dWj(t)
dt, t ∈ R, Yj(0) = 0, (1)
where W (t) = (W1(t), . . . ,Wn(t))′ , t ∈ R , is an Rn -valued standard Brownianmotion, the derivatives dYj(t)/dt and dWj(t)/dt are in the random distributionsense, and pj ’s and qj ’s are constants such that
0 < qj <∞, −qj < pj <∞, j = 1, . . . , n. (2)
The two parameters pj and qj describe the memory of Yj(t) . Let (Ft)t≥0 be theaugmentation of the filtration generated by the process (Y (t))t≥0 , which is the un-derlying information structure of M . The process (Y (t))t≥0 is a semimartingalewith respect to (Ft) with the following explicit semimartingale representations:
Yj(t) = Bj(t)−Z t
0
»Z s
0
pj(2qj + pj)(2qj + pj)e
qju − pje−qju
(2qj + pj)2eqjs − p2je−qjs
dYj(u)
–ds,
Yj(t) = Bj(t)−Z t
0
»Z s
0
pj(2qj + pj)e−(pj+qj)(s−u) (2qj + pj)e
2qju − pj(2qj + pj)2e2qju − p2
j
dBj(u)
–ds,
where (Bj(t))t≥0 is the so-called innovation process, i.e., an R -valued standardBrownian motion such that σ(Yj(s) : 0 ≤ s ≤ t) = σ(Bj(s) : 0 ≤ s ≤ t) for t ≥ 0 .
The market model M consists of n risky and one riskless assets. The priceof the riskless asset is denoted by S0(t) and that of the ith risky asset by Si(t) .We put S(t) = (S1(t), . . . , Sn(t))′ . The dynamics of the Rn -valued process S(t)are described by the stochastic differential equation
dSi(t) = Si(t)hµi(t)dt+
Xn
j=1σij(t)dYj(t)
i, t ≥ 0,
Si(0) = si, i = 1, . . . , n,(3)
while those of S0(t) by the ordinary differential equation
dS0(t) = r(t)S0(t)dt, t ≥ 0, S0(0) = 1, (4)
where the coefficients r(t) ≥ 0 , µi(t) , and σij(t) are continuous deterministicfunctions on [0,∞) and the initial prices si are positive constants. We assumethat the n× n volatility matrix σ(t) = (σij(t))1≤i,j≤n is nonsingular for t ≥ 0 .
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For the market model M , we consider an agent with initial endowmentx ∈ (0,∞) who invests, at each time t , πi(t)Xx,π(t) dollars in the ith risky assetfor i = 1, . . . , n and [1 −
Pni=1 πi(t)]X
x,π(t) dollars in the riskless asset, whereXx,π(t) denotes the agent’s wealth at time t . Here, we choose the self-financingstrategy π(t) = (π1(t), . . . , πn(t))′ from
AT :=
(π = (π(t))0≤t≤T :
π is an Rn-valued, progressively measurable
process satisfyingR T0‖π(t)‖2dt <∞ a.s.
)or
A := (π(t))t≥0 : (π(t))0≤t≤T ∈ AT for every T ∈ (0,∞) .Let α ∈ (−∞, 1) \ 0 and c ∈ R . We explicitly solve the following three
optimal investment problems for M :
V (T, α) := supπ∈AT
1
αE [(Xx,π(T ))α] , (P1)
J(α) := supπ∈A
lim supT→∞
1
αTlogE [(Xx,π(T ))α] , (P2)
I(c) := supπ∈A
lim supT→∞
1
TlogP
hXx,π(T ) ≥ ecT
i. (P3)
Problem P1 is Merton’s classical portfolio optimization problem. Problem P2 isthe maximization of growth rate of expected utility of wealth over the infinitehorizon. Problem P3 is the maximization of the large deviation probability thatthe wealth grows at a higher rate than a given benchmark.
The estimation of paremeters from real market data is also considered.
References
[1] V. Anh and A. Inoue, Financial markets with memory I: Dynamic models.Stoch. Anal. Appl., 23:275–300, 2005.
[2] V. Anh, A. Inoue and Y. Kasahara. Financial markets with memory II: In-novation processes and expected utility maximization. Stoch. Anal. Appl.,23:301–328, 2005.
[3] A. Inoue and Y. Nakano. Optimal long term investment model with memory,submitted.Available at http://www.math.hokudai.ac.jp/˜inoue/oltimm.pdf.
[4] A. Inoue, Y. Nakano and V. Anh. Linear filtering of systems with memoryand application to finance. J. Appl. Math. Stoch. Anal., to appear.
Department of Mathematics, Faculty of Science, Hokkaido University,Sapporo 060-0810, Japane-mail: [email protected]
145
Martingale methods of estimation in thenonlinear autoregression model
Dmytro Ivanenko, Ukraine
We consider the difference equation
ξk = f(ξk−1, a) + εk, k ∈ N,
were ξ0 is a prescribed random values, f is a prescribed function, a is an unknownscalar parameter and (εk) is a square integrable difference martingale with respectto some flow (Fk, k ∈ Z+) of σ -algebras such that the random variable ξ0 is F0 -measurable.
Define
Fn(a) =1
n
nXi=1
f(ξi−1, a)h(ξi−1).
Here h is an F0 -measurable random function.The converges in distribution of the normalized by
√n deviation of the esti-
mator
an = F−1n
1
n
nXi=1
ξiξi−1
!from the true value of the parameter a under the condition
limN→∞
limn→∞
1
n
nXk=1
Eε2kI|εk| > N = 0
and additional technical assumptions on f is proved.The use of stochastic calculus underlying our approach allows to dispense with
the assumptions of ergodicity and even asymptotic stationarity of the sequence(εk) , thereat the limiting distribution of the studied statistic may be other thannormal.
Let now the sequence (εk) be ergodic. Then the distribution of√n(an − a)
will be asymptotically normal. In the model
ξk = af(ξk−1) + εk,
we obtain the explicit expression for the function h minimizing its variance.
Department of RadiophisicsKyiv University,64 Volodymyrska Str.e-mail: [email protected]
146
Consistency of M-estimates in generalnonlinear regression models
Alexander V. Ivanov and Igor V. Orlovsky, Ukraine
Consider a regression model
X(t) = g(y(t), θ) + ε(t), t ≥ 0 ,
where g(y, θ) is a non random function defined on Y × Θc , Θc is a closure inRq of an open set Θ , Y ⊂ Rm is a compact region of regression experimentdesign. Borel function y(t) : [0,∞) → Y is a design of regression experiment,θ = (θ1, ..., θq) ∈ Θc is an unknown parameter, ε(t), t ∈ R1 is a real valuedmean-square continuous measurable stationary process with zero mean. We donot assume function g(y, θ) to be a linear form of coordinates of the vector θ .
The problem of nonlinear regression model parameter estimating is an impor-tant problem of statistics of random processes. For the estimating of unknownparameters, so called, M -estimates are widely used. The most known estimatesin this class are least square estimates (l.s.e.) and least moduli estimates (l.m.e.).
Definition. M -estimate of unknown parameter θ obtained by the observationsX(t), t ∈ [0, T ) is said to be any random vector bθT that minimizes in τ ∈ Θc
the integral functional MT (τ) =1
T
TR0
ρ(X(t)− g(y(t), τ))dt with countinuous risk
function ρ : R1 → R1 .
Consistency of M -estimates for nonlinear regression model with independentidentically distributed observation errors is considered in [1]. Some facts on con-sistency of l.s.e and l.m.e. can be found in [2].
Sufficient conditions for strong consistency of M -estimates of an unknown pa-rameter θ of nonlinear regression model with random noise that satisfies weak andlong-range dependece conditions are presented in this talk. This results generalizesome statements of the paper [1].
Assume that1) ε(t), t ∈ R1 satisfy weak dependence condition, i.e. for some δ > 0
E|ε(0)|2+δ <∞ and∞Z0
(α(r))δ
2+δ dr <∞,
whereα(r) = sup
A∈σ(−∞,s), B∈σ(s+r,∞)
|P (AB)− P (A)P (B)|,
σ(a, b) is σ -algebra generated by the random variablesε(t), t ∈ (a, b) .
147
2) ε(0) is a symmetric and unimodal random variable.
Suppose B is σ -algebra of Borel subsets of Y . For any A ∈ B set
µT (A) = T−1m(t ∈ [0, T ] : y(t) ∈ A),
where m is Lebesgue measure on [0,∞) .3) Measures µT converges weakly to µ as T → ∞ and for any ε > 0
µ(y ∈ Y : g(y, θ)− g(y, τ) = 0) < 1 for any τ /∈ vθ(ε) = τ ∈ Rq : ‖τ − θ‖ < ε .
4) ρ(x) is continuous even unimodal with mode in zero function, ρ(0) = 0 .
5) There exists κ > 0 such that for any x1, x2 ∈ R1
|ρ(x1)− ρ(x2)| ≤ κ|x1 − x2| .
6)∞R0
[P|ε(0)| < z − P|ε(0)− b| < z] dρ(z) > 0 for any b > 0 .
Theorem 1 If the conditions 1) - 6) are fulfilled, then bθT → θ a.s. as T →∞ .
Instead of the conditions 1) and 2) let us introduce the long-range dependencecondition
7) ε(t), t ∈ R1 is a Gaussian random process with covariance functionB(t) = Eε(0)ε(t) = L(|t|)
|t|α , 0 < α < 1 , B(0) = 1 , where L(t) is a slowly varyingfunction at infinity.
Theorem 2 If the conditions 3) - 7) are fulfilled, then bθT → θ a.s. as T →∞ .
References
1. F. Liese and I. Vajda Consistency of M -estimates in general regressionmodels. Journal of Multivariate Analysis, 1994, 50, 1, p. 93-114.
2. A. V. Ivanov Asymptotic Theory of Nonlinear Regression. Kluwer AP,1997.
[Alexander V. Ivanov]National technical universityof Ukraine "KPI"Peremogi avenue 37, Kieve-mail: [email protected]
[Igor V. Orlovsky]National technical universityof Ukraine "KPI"Peremogi avenue 37, Kieve-mail: [email protected]
148
Exit from the interval by the difference of tworenewal processes
V.F. Kadankov, Ukraine
Let η, ξ, κ, δ ∈ (0,∞) be random variables. Denote by F (x) = P [ η ≤ x ],G(y) = P [ ξ ≤ y ] cumulative distribution functions of the random variablesη, ξ. Introduce the sequences η, ηn, ξ, ξn, n ∈ N of independent identicallydistributed variables and for x, y ≥ 0 define independent random sequences
η0(x) = 0, η1(x) = ηx, ηn+1(x) = ηx + η1 + · · ·+ ηn, n ∈ N,ξ0(y) = 0, ξ1(y) = ξy, ξn+1(y) = ξy + ξ1 + · · ·+ ξn, n ∈ N, (1)
where ηx, ξy are the random variables defined by their cumulative dis-tribution functions: P [ ηx ≤ u ] = [F (x + u) − F (x)](1 − F (x))−1,P [ ξy ≤ v ] = [G(y + v)−G(y)](1−G(y))−1, u, v ≥ 0.
For all x, y ≥ 0 define independent renewal processes αy(t), βx(t) gener-ated by the random sequences (1): αy(t) = max n ∈ N ∪ 0 : ξn(y) ≤ t ,βx(t) = max n ∈ N ∪ 0 : ηn(x) ≤ t . Introduce monotone independent randomwalks generated by the random variables κ, δ : κ0 = 0, κn = κ′1+· · ·+κ′n; δ0 = 0,δn = δ′1 + · · ·+ δ′n, where κ, κ′n, δ, δ′n, n ∈ N are the sequences of the inde-pendent identically distributed random variables. Define a right-continuous stepprocess
Dxy(t)t≥0 = καy(t) − δβx(t) ∈ R, x, y ≥ 0; Dxy(0) = 0.
We will call the process Dxy(t) t≥0 the difference of two renewal processes. Forall t ≥ 0 introduce right-continuous linear components:
η+x (t) =
t+ x, 0 ≤ t < ηx,t− ηβx(t)(x), t ≥ ηx
∈ R+, x ≥ 0,
ξ+y (t) =
t+ y, 0 ≤ t < ξy,t− ξαy(t)(y), t ≥ ξy
∈ R+, y ≥ 0.
Define a Markov process accompanying the process Dxy(t)t≥0 :
Xtxy =
˘Dxy(t), η+
x (t), ξ+y (t)
¯t≥0
∈ R× R2+, X0
xy = 0, x, y, x, y ≥ 0.
For k ∈ R+ define the random variables
τkxy = inf t : Dxy(t) > k , T kxy = Dxy(τkxy)− k, ηkxy = η+x (τkxy)
τxyk = inf t : Dxy(t) < −k , T xyk = −Dxy(τxyk )− k, ξxyk = ξ+y (τxyk )
Let us fix B B ≥ 0, and let k ∈ [0, B], r = B−k, X0xy = 0, x, y, x, y ≥ 0. Intro-
duce a random variable χ = inf t : Dxy(t) /∈ [−r, k] the first exit time from theinterval [−r, k] by the process Dxy(t)t≥0 . Denote by A k = Dxy(χ) > k the
149
event that the process exits through the upper level, and Ar = Dxy(χ) < −r the event that exit occurs through the lower level. Define
T = (Dxy(χ)− k) IA k + (−Dxy(χ)− r) IAr , L = η+x (χ) IA k + ξ+
y (χ) IAr
the value of the overshoot of the interval [−r, k] by the process Dxy(t)t≥0 andthe value of the linear component at the epoch of the exit. Introduce the notation:
fkxy(du, dl, s) = Ehe−sτ
kxy ;T kxy ∈ du, ηkxy ∈ dl
i,
fxyk (du, dl, s) = Ehe−sτ
xyk ;T xyk ∈ du, ξxyk ∈ dl
i,
F+xy(du, dl, s) = fkxy(du, dl, s)−
ZZR2+
fxyr (dv, dν, s) fv+B0ν (du, dl, s),
F−xy(du, dl, s) = fxyr (du, dl, s)−ZZ
R2+
fkxy(dv, dν, s) fν0v+B(du, dl, s).
Теорема 1 The Laplace transforms of the joint distribution of χ, T, L thefirst exit time from the interval [−r, k] by the process Dxy(t)t≥0, the value ofthe overshoot through the boundary and the value of a linear component at theepoch of the exit satisfy the formulae
E[e−sχ;T ∈ du, L ∈ dl, Ak] = F+xy(du, dl, s) +
ZZR2+
F+xy(dv, dν, s) K+
vν(du, dl, s),
Eˆe−sχ;T ∈ du, L ∈ dl, Ar
˜= F−xy(du, dl, s) +
ZZR2+
F−xy(dv, dν, s) K−vν(du, dl, s),
where K±vν(du, dl, s) =
∞Pn=1
K±vν(du, dl, s)∗(n) is the series of iterations,
K±vν(du, dl, s)∗(1)
def= K±
vν(du, dl, s),
K±vν(du, dl, s)∗(n+1) =
ZZR2+
K±vν(du1, dl1, s)K
±u1l1
(du, dl, s)∗(n), n ∈ N
are the iterations of the kernels K±vν(du, dl, s), which are defined by the formulae
K+vν(du, dl, s) =
ZZR2+
fν0v+B(du1, dl1, s) fu1+B0l1
(du, dl, s),
K−vν(du, dl, s) =
ZZR2+
fv+B0ν (du1, dl1, s) fl10u1+B(du, dl, s).
References
[1] V.F. Kadankov. (2006) Exit from an interval by a difference of two renewalprocesses. Theory of Stochastic Processes (in print).
Institute of Mathematics of Ukrainian National Academy of Sciences,3 Tereshchenkivska st. 01601, Kyiv-4, Ukrainee-mail: [email protected]
150
Total duration of stay inside the interval byPoisson process with negative exponential
component
T.V. Kadankova, Ukraine
Let η ∈ (0,∞) be a positive random variable and γ be an exponentially dis-tributed random variable with parameter λ > 0 i.e. P [ γ > x ] = e−λx, x ≥ 0.Introduce a random variable ξ ∈ R by its cumulative distribution function
F (x) = q exλ Ix≤0 + (q + (1− q)P [ η < x ]) Ix>0, q ∈ (0, 1).
Define a compound Poisson process ξ(t); t ≥ 0 with the cumulant
k(p) = c
Z ∞
−∞(e−xp − 1) dF (x) = a1
p
λ− p+ a2(E e−pη − 1), < p = 0,
where c > 0, a1 = qc, a2 = (1 − q)c. Observe, that jumps of the processξ(t); t ≥ 0 occur with intensity c; with probability 1 − q process has posi-tive jumps of value η,and with probability q there occur negative jumps whichare exponentially distributed with parameter γ.
Define the resolvent function Rs(x), x ≥ 0 of a Poisson process with a negativeexponential component [1] by the following formula:
Rs(x) =1
2πi
Z α+i∞
α−i∞exp R(p, s) dp, α > c(s),
where c(s) ∈ (0, λ) is a unique root [2] of the equation k(p)− s = 0, s > 0 in thesemi-plane < p > 0, and
R(p, s)−1 = a1p+ (p− λ)[s− a2(E e−pη − 1)], < p ≥ 0, p 6= c(s).
Let y ≥ 0, ξ(0) = 0, and τy = inf t : y + ξ(t) < 0 , σy = σy(τy) the first exittime from the upper semi-plane by the process y + ξ(·) and the total duration ofstay inside the interval [0, B] up to the instant τy.
Theorem 1 ([1]) Let ξ(t); t ≥ 0, ξ(0) = 0 be a Poisson process with a negativeexponential component. Then the integral transform Ds
a(y) = E[exp−sτy−aσy ],y ≥ 0, a ≥ 0 of the joint distribution of τy, σy for s > 0 satisfy the equality
Dsa(y) =
hV sa (B − y)− aRs+a(B − y) e−c(s)(B−y)
iV sa (B)−1E e−sτy , y ≥ 0,
where Rs(x)def= 0 for x < 0, and V sa (x), x ∈ R is given by the following identity
V sa (x) = 1 + a(λ− c(s))
Z x
0
e−uc(s)Rs+a(u) du, x ≥ 0; V sa (x) = 1, x < 0.
151
Theorem 2 ([1]) Let ξ(t); t ≥ 0, ξ(0) = 0 be a Poisson process with a nega-tive exponential component, B > 0, a ≥ 0 and for y ∈ R
σy(t) =
Z t
0
I y + ξ(u) ∈ [0, B] du, C sa (y) = s
Z ∞
0
e−stE [e−aσy(t)] dt
be the total duration of stay inside the interval [0, B] by the process y+ξ(·) on thetime interval [0, t] and the integral transform of σy(t). Then the function C s
a (y),y ∈ R for s > 0 satisfies the following formulae
C sa (y) = v sa (B − y)− aRs+a(B − y) +D s
a (y)C∗(B), y ≥ 0,
C sa (−y) = 1− E [e−sτ
y
] +
Z ∞
0
E [ e−sτy
; T y ∈ du ]C sa (u), y > 0,
where τy = inft : ξ(t) > y, T y = ξ(τy)− y, y ≥ 0,and
v sa (x) = 1 + aλ
Z x
0
Rs+a(u) du, x ≥ 0, v sa (x) = 1, x < 0,
C∗(B) =
aλc(s)
“v sa (B)− V s
a (B)ec(s)B”V sa (B)
r(c(s), s) + a(λ− c(s))BR0
(V sa (x)− a e−xc(s)Rs+a(x)) dx
r(c(s), s) =d
dpR(p, s)−1
˛p=c(s)
.
The joint distribution of the first exit time from the interval and the valueof the overshoot through the boundary at this epoch has been used to derivethe formulae of the theorems. The aforementioned distribution has been obtainedfor a Levy process with the cumulant of general form in [3], and for the Poissonprocess with a negative exponential component in [1].
References
[1] Kadankov V. F., Kadankova T. V. (2006). Two-boundary problems ofthe Poisson process with a negative exponential component 58(5).
[2] Borovkov A.A. (1976). Stochastic processes in queueing theory. SpringerVerlag.
[3] Kadankov V. F., Kadankova T. V. (2005). On the distribution of thefirst exit time from an interval and value of overjump through the bordersfor processes with independent increments. Ukr. Math. J. 57(10), 1359-1384.
Mathematical Statistics, Center for Statistics, Hasselt University,Universitaire Campus, Building D, B-3590 Diepenbeek, Belgiume-mail: [email protected]
152
The time of ruin for the modified risk processin a Markov environment
Eugene Karnaukh, Ukraine
Two-sided boundary problems for the semi-continuous processes, defined on afinite Markov chain were considered in papers [1], [2]. We can use results of thesepapers for investigation of the modified risk process (analog of the processes, whichwere considered in [3], [4]).
Let x(t) be a finite irreducible ergodic Markov chain with state spaceE′ = 1, . . . ,m and transition matrix P(t) = exptQ . Lets construct processξ(t) in the following way. ξ(0) = 0 . If x(t) = k , k = 1, ...,m , then the incrementsof ξ(t) coincide with the increments of the process
ξk(t) = akt−X
n≤εk(t)
ξkn,
where ak > 0 , εk(t) are the Poisson processes with the rates λk . ξkn are in-dependent positive random variables with the bounded expectations mk . Theprocess Z(t) = ξ(t), x(t) is a upper semi-continuous process, defined on thefinite Markov chain (see [1]) with the cumulant function
Ψ(α) = iαA +
Z 0
−∞(eıαx − I) Π(dx) + Q,
where A = ‖δkrak‖ , Π(dx) =‚‚δkrλkdP
˘−ξkn < x
¯‚‚ , x < 0 .The process Z(t) has the next interpretation: x(t) describes the environmental
conditions and ξ(t) describes the reserves of company with no initial capital.Further we assume that the initial capital of the company is equal u > 0 . If thecapital becomes equal b > u at some moment T1 , then all income are paid asdividends until the claim arrive(the moment T2 ), then we have again the reserveprocess only with the initial capital b− ξ , where ξ is the claim size. Namely, weconsider the process ηu,b(t) , which is determined by the next stochastic relations
ηkru,b(t).=
8><>:u+ ξkr(t) t < T1,b t ∈ (T1, T2), T1 <∞,
ηjrb−ξj
1,b(t− T2) t > T2, x(T2) = j, T1 <∞,
(1)
where indices kr mean that x(t) = r, x(0) = k . Note that T2.= T1 + ζ∗ , ζ∗ is the
time of the first claim, ζ∗ is independent of T1 .The process ηu,b(t) is called the risk reserve process in a Markov environment
with bounded reserves.Denote the first exit time from interval τ(u, b) = inf t ≥ 0 : ξ(t) + u /∈ [0, b] andtransforms B(s, u, b) = E e−sτ(u,b) , Bb(s, u) = E
he−sτ(u,b), ξ(τ(u, b)) + u ≥ b
i,
153
Bb(u) = P ξ(τ(u, b)) + u ≥ b , Bb(s, u) = Ehe−sτ(u,b), ξ(τ(u, b)) + u ≤ 0
i,
B∗(s, b) =R 0
−b dF0(z)Bb(s, b + z) . Note that the representations of these trans-forms in terms of the resolvent function were obtained in [1] and [2].
Denote the time of ruin
θ(u, b) = inf t > 0 : ηu,b(t) ≤ 0 .
Theorem 1 The distribution of the ruin time θ(u, b) is determined by the gen-erating function
Ee−sθ(u,b) = B(s, u, b)− sBb(s, u) (sI + Λ−Q)−1 P−1s −
−Bb(s, u) (sI + Λ−Q−ΛB∗(s, b))−1×
ׄZ 0
−bΠ(dz) (I−B(s, b+ z, b))−ΛB∗(s, b) (sI + Λ−Q)−1 P−1
s
«.
For the expectation we have
(Eiθ(u, b))′ = (Eiτ(u, b))′ + Bb(u)
„I−
Z 0
−bΠ(dz)Bb(b+ z) (Λ−Q)−1
«−1
×
ׄe′ +
Z 0
−bΠ(dz) (Eiτ(b+ z, b))′
«,
where Ei(·) is the expectation under condition x(0) = i , e′ = (1, . . . , 1)′ .
References
[1] D.V. Gusak, Boundary problems for the processes with independent incrementson the Markov chains and semi-Markov processes, Inst. of Math. of Ukr.Acad.of Science, Kiev, 1998, pp. 320 (in ukranian).
[2] V.S. Korolyuk, V.M. Shurenkov The method of resolvent in the boundary prob-lems for the random walks on a Markv chain, Ukr.Math.Journal, 29 (1977),no. 4, pp. 464-471 (in russian).
[3] D.V. Gusak, On modifications for the risk processes, Teor. Jmovirn. Mat. Stat.56 (1997), pp. 87-95 (in ukrainian).
[4] N.V. Kartashov, On ruin probability for a risk process with bounded reserves.Teor. Jmovirn. Mat. Stat. 60 (1999), pp. 46-58 (in ukrainian).
Department of Probability Theory and Mathematical Statistics,Faculty of Mechanics and Mathematics,Kyiv National Taras Shevchenko University,Volodymyrska 64, 01033, Kyiv, Ukrainee-mail: [email protected]
154
The asymptotic of the ruin probability for thestate-nonhomogeneous risk process
Kartashov M.V., Stroyev O.M., Ukraine
1. Let c(x), x ∈ R+ is the measurable positive function such that 1/c(x)
is locally integrable. Let also Z(t) =Pν(t)k=1 ξk be a compound Poisson process
with independent identically distributed payment values (ξn, n ≥ 1) and withindependent Poisson process ν(t).
Consider the right-continuous Markov process, that satisfies the stochasticequation
dXt = c(Xt)dt− dZt, t ≥ 0, X0 = x. (1)
with state-dependent premium rate c(x).Define the ruin probability
q(x) = P (∪ t≥0 Xt < 0 | X0 = x), x ≥ 0. (2)
Introduce following notation
λ = Mν(1), m = Mξ1, F (x) = P (ξ1 < x). (3)
Assume that the Cramer’s condition is satisfied:
γ ≡ sup(s ≥ 0 : M exp(sξ1) <∞) > 0. (4)
Under (4) for all Res < γ is correctly defined the analytical moments gener-ating function
bf(s) =
Z ∞
0
exp(sx)dF (x), bg(s) = ( bf(s)− 1)/s. (5)
Assume also that the premium rate satisfies for some c > 0 the condition ofthe exponential convergence:
∃β > 0 : c(x)− c = O(exp(−βx)), x→∞, (6)
and simultaneously the balance condition is true:
c > λm. (7)
Consider the Lundberg’s exponent for the risk process with the constant premiumrate c :
α ≡ sup(s < γ : λ bf(s)− λ < cs) ∈ (0,∞). (8)
155
From the continuity of bf(s) on the interval [0, γ) it follows that under the Lund-berg condition: α < γ the exponent α is the unique positive solution of theequation:
cα− λ bf(α) + λ = 0. (9)
2. Following assertion is a generalization of the Lundberg’s estimate for theruin probability, since for c(x) ≡ c it implies this estimate.
Theorem 1. Let the Markov process (Xt, t ≥ 0) is given from the equation(1), the Cramer’s condition (4) holds, and for some c > 0 the representation (6)as well as the balance condition (7) are valid.
Define the exponents α ≤ γ from (8) and (4).(a) If α < γ then there exists ρ ∈ (α, γ) and the constant Cα such that
holds following representation for the ruin probability:
q(x)− Cα exp(−αx) = O(exp(−ρx)), x→∞, (10)
and for all sufficiently small perturbations c(x) − c the constant Cα is positive,so that there exist the limits
lim x→∞ x−1 ln q(x) = −α, Cα = lim x→∞ exp(αx)q(x) > 0. (11)
(б) If α = γ then
q(x) = O(exp(−βx)), x→∞, ∀β < γ. (12)
Further the representations above are updated to the exact inequalities interms of such metrics for the perturbation of the premium rate:
ρ±s = sup x≥0(exp(sx)− 1)(c(x)− c)±,
σ±s =
Z ∞
0
(exp(sx)− 1)(c(x)− c)±dx, (13)
where y± – is the positive and negative parts of y.These inequalities turn to exact (Lundberg’s) equalities for the constant Cα
for the case when c(x) ≡ x, since in this case ρ±s = σ±s = 0.The details and examples one can find in [1].
References
[1] Карташов М.В., Строєв О.М., Наближення Лундберга для функцiїризику у майже однорiдному середовищi, Теорiя ймовiрностей i математичнастатистика, 2005, в.73, с.64-72.
Kiyv National Taras Shevchenko University,Mech-Math Faculty, Kiyv, Volodymirska 64e-mail: [email protected]
156
Invariance principle for additive functionals ofrandom walks
Iurii Kartashov, Alexey Kulik, Ukraine
We propose a new approach to studying the limit behavior of the families ofadditive functionals of the processes, which converge to some Markov process X .All the processes below are defined on [0, 1] and have the same locally compactphase space (X , ρ) .
Definition 1 The sequence of the processes Xn provides the Markov approx-imation for the Markov process X , if for every γ > 0 there exist a constant kγand sequence of two-component processes Yn = (Xn, X) , defined on anotherprobability space, such that
(i) Xnd=Xn, X
d=X ;
(ii) the process Yn possess the Markov property at the points ikγ
n, i ≤ n
kγ,
w.r.t. the filtration Fnt = σ(Yn(s), s ≤ t) ;(iii) lim supn→+∞ P
“supi ρ(Xn(
ikγ
n), X(
ikγ
n)) > γ
”< γ.
Our general result is the analogue of the E.B.Dynkin’s theorem on a conver-gence of W -functionals. We consider the sequence of the additive functionals ofthe type
φt,ns =X
i:s≤ in<t
Fn“Xn(
i
n), . . . , Xn(
i+ L
n)”, 0 ≤ s ≤ t ≤ 1.
For such functionals we define their characteristics fn byf t,ns (x) = E[φt,ns |Xn(0) = x] . We suppose that ‖Fn‖∞ → 0, n→ +∞ .
Theorem 1 Suppose that1) the sequence of the processes Xn provides the Markov approximation for
the Markov process X ;2) the sequence fn converges uniformly to a jointly continuous function f ,
which is the characteristics of some W -functional φ of the process X .Then φt,ns ⇒ φts, 0 ≤ s ≤ t ≤ 1, n→ +∞ .
We show that in the typical situations, including one-dimensional randomwalks with the stable domain of attraction and multidimensional random walkswith finite second moment, the random walk provides the Markov approximationfor the limiting process with independent increments. As a corollary, we obtainthe invariance principle for additive functionals of random walks.
Institute of Mathematics,Ukrainian National Academy of Sciences,Kiev 01601 Tereshchenkivska str. 3e-mail: [email protected]
157
On robustness of multivariate Bayesianforecasting under functional distortions of
priorsAlexey Kharin, Pavel Shlyk, Belarus
Bayesian approach is efficiently used to forecast a vector of dependent compo-nents, especially in the case of small samples [3]. Bayesian approach is based onprior information on an object being observed. As in practice prior informationis usually misspecified, robustness analysis is necessary for correct use and forconstruction of Bayesian predictors.
Suppose that the vector of observations x = (xt)Tt=1 ∈ X ⊆ Rn×T stochasti-
cally depends on θ with the hypothetical conditional probability density function(p.d.f.) p0(x|θ) , where θ ∈ Θ ⊆ Rm is an unobserved vector of model parame-ters with the hypothetical p.d.f. π0(θ) . The problem is to forecast y ∈ Y ⊆ Rn
that stochastically depends on x and θ with the hypothetical conditional p.d.f.g0(y|x, θ) .
To formalize uncertainty of prior information, we consider the follow-ing admissible distortions of the hypothetical model. Suppose θ has a p.d.f.π(θ) ∈ Πε
`π0(·)
´, where Πε
`π0(·)
´is a set of admissible distorted p.d.f.s of
θ , defined using Lp -norm [1]:
Πε
`π0(·)
´=nπu(θ) : ‖ u(·) ‖Lp
≤ εo, ε > 0, u(·) : Rm → [0,∞),
πu(θ) =πu(θ)R
Θπu(θ)dθ
, πu(θ) =
( “`π0(θ)
´1/p+ 1
pu(θ)
”p, 1 ≤ p <∞,
π0(θ)eu(θ), p = ∞.
The upper risk functional is constructed and the analytic expression for therobust prediction statistic is obtained for the distortion described above. Asymp-totic properties for the latter have been investigated. Analogous results have beenobtained for the case of distortions defined using the weighted C-metric [2]. Sim-ulations have been carried out for the both cases.
References
[1] P.Gustafson, Local Sensivity of Posterior Expectations, Ph.D. Dissertation,Carnegie Mellon University, Pittsburgh, 1994.
[2] A.Kharin Minimax Robustness of Bayesian Forecasting under Functional Dis-tortions of Probability Densities, Austrian Journal of Statistics. 2002. Vol. 31,p. 177-188.
[3] M.West, J.Harrison, Bayesian Forecasting and Dynamic Models, Springer,New York, 1989.
Department of Probability Theory and Mathematical StatisticsBelarusian State Unversity,Independence avenue 4, 220030 Minsk, Belaruse-mail: [email protected], [email protected]
158
Statistical analysis of high-order Markovchains
Yuriy Kharin, Belarus
Mathematical modeling of complex systems and processes in genetics, eco-nomics, sociology and engineering needs adequate probability models for discretetime series xt ∈ A , t ∈ N (A = 0, 1, . . . , N−1 is the finite set with 2 ≤ N <∞elements) with "long memory" [1]-[3]. A well known model for these discrete timeseries is the Markov chain of high order s ∈ N defining the "memory depth" [4].Unfortunately, the number of parameters Ds = Ns(N − 1) for this model in-creases exponentially w.r.t. the order s , and statistical inferences need to observea realization x1, . . . , xn of inadmissibly large size n > Ds .
This situation generates a very topical problem of construction and statisticalanalysis of "low-parametric" high-order Markov chains. A. Raftery proposed [3] ahigh-order Markov chains as a mixture of s simple Markov chains of first order.Here we propose and analyze a new model - high-order Markov chain with partialconnections MC(s ,r ) [5].
Let xt ∈ A , t ∈ N be a Markov chain of the order s defined on the probabilityspace (Ω,F ,P) ; r ∈ 1, . . . , s be a parameter called the number of connections;M0r = (m0
1, . . . ,m0r) ∈M be an integer r -vector with ordered components
1 = m01 < m0
2 < . . . < m0r ≤ s
called the pattern of connections; P = (pi1,...,is,is+1) , i1, . . . , is+1 ∈ A be an(s+ 1) -dimensional matrix of one-step transition probabilities:
pi1,...,is,is+1 = Pxt = is+1 | xt−1 = is, . . . , xt−s = i1, t > s;
Q0 = (q0j1,...,jr,jr+1) , j1, . . . , jr+1 ∈ A be an (r+1) -dimensional stochastic matrix.We introduce the MC(s ,r )-model by the special form of the transition probabil-ities:
pi1,...,is,is+1 = q0im0
1,...,i
m0r,is+1 , i1, . . . , is+1 ∈ A;
it means that the probability of transition to the state is+1 at the time mo-ment t depends not on all former states i1, . . . , is of the process but only on rselected states im0
1, . . . , im0
r. If r = s , M0
r = (1, . . . , s) , then MC(s ,s) is the“classical” Markov chain of the order s . The model MC(s ,r ) is described byds = Nr(N − 1) + r − 1 parameters Q0 , M0
r .The paper presents the following results:
• a review of "low-parametric" models for discrete time series with "longmemory";
• probabilistic properties of the MC(s ,r )-model;
• statistical estimators of the parameters M0r , Q0 , r and their asymptotic
properties;
159
• statistical tests on parameters M0r , Q0 and their asymptotic properties;
• estimates for the computer complexity of proposed algorithms of statisticalanalysis of the MC(s ,r );
• results of numerical experiments.
References
[1] D. Collet. Modeling of binary data. - Chapman & Hall, 2003, 252p.
[2] M. Waterman. Mathematical methods for analysis of DNA sequences. - CRCPress, 1999, 230p.
[3] A.E. Raftery. A model for high-order Markov chains. - "Journal of Royal Stat.Soc.", 1985, Vol.47, No. 3, pp. 40-44.
[4] J.L. Doob. Stochastic Processes. - Wiley, 1953, 516p.
[5] Yu. Kharin. Markov chains with r -partial connections and their statisticalestimation. - "Doklady of National Academy of Sciences of Belarus", 2004,Vol. 48, No. 1, pp. 40-44.
Department of Mathematical Modeling and Data AnalysisBelarusian State Unversity,4 Independence av., 220030 Minsk, Belaruse-mail: [email protected]
A nonlinear filtering of smooth signals
Rafail Khasminskii, USA
A nonlinear online Kalman type filter is proposed for the estimation of un-known function with the known smoothness for the diffusion observed processwith small noise and the drift coefficient depending on the unknown signal. TheLyapunov’s functions method is proposed for the analysis of this filter. Under someconditions the filter with the best rate of convergence risks to zero, as intensity ofnoise tends to zero, is found.
Wayne State University,Detroit, USA
160
Identification of continuous variabledistribution by means of numeric characteristic
Kirichenko L., Shklovets A., Ukraine
In the paper some functional characteristic for distribution of continuous vari-able (CV) ξ suggested. Let CV ξ be presented by a sample of the size n from itspopulation; a variable Rξ(n) be the swing of the sample. Then the function
Idξ(n) =(M [Rξ(n)])2
D[ξ],
appears to be numerical characteristic of distribution, where M [Rξ(n)] is anexpectation of sample swing, D[ξ] is variance of CV ξ .
The values of function Idξ(n) determined with the distribution function Fξ ,taking into account, that
M [Rξ(n)] =
Z +∞
−∞(1− Fnξ (x)− (1− Fξ(x))2)dx.
Two variables belong the same distribution type, if they can be found one fromanother as a linear transformation. In the paper we show, that function Idξ(n)has the same value for all CV from the same distribution type.
The function Idξ(n) could be used to identify the distribution type by datasample. The estimate for Idξ(n) , found on k samples of the size n , is the value
Id(n) =
`1k
Pki=1(Xi
max(n)−Ximin(n)
´21k
Pki=1 S
2i
,
where Ximin and Xi
max are maximal and minimal values of the i -th sample, Siis standard deviation of the i -th sample. In the paper it is shown, that Id(n) isconsistent estimate.
The research of behavior of Id(n) for different distributions was carried out.To identify the distribution type by data sample, it was suggested the use ofstandard deviation of experimental curve from the theoretical one.
Kharkov National University of Radioelectronics61166, Lenin ave. 14, Kharkov, Ukrainee-mail: [email protected]
161
Some properties of weight functions inTauberian theorems.
B. Klykavka, A. Olenko, Ukraine
Abelian and Tauberian theorems find numerous applications in obtaining differ-ent asymptotical properties of random processes and fields. The majority resultsof this type are connected with asymptotical behavior of spectral or correlationcharacteristics on the infinity or in zero.
Let Rn - Eucledian space, ξ(t), t ∈ Rn – real valued continuous homogeneousand isotropic random field with zero mean. It is known that there exists suchbounded nondecreasing function Φ(x), x ≥ 0 (which is called spectral function ofthe field), that correlation function Bn(r) admits such reprsentation
Bn(r) = Bn(‖t‖) = Eξ(0)ξ(t) = 2n−2
2 Γ“n
2
”Z ∞
0
Jn−22
(rx)
(rx)n−2
2
dΦ(x),
where Jν(z) – Bessel function of the first type, ν > − 12.
Denoting
ba(r) = (2π)nZ ∞
0
J2n2
(rx)
(rx)ndΦa(x), Φa(x) :=
Φ(a+ x)− Φ(a− x), 0 ≤ x < a;Φ(a+ x), x ≥ a,
it was shown in [1], that ba(r) Φ“a+ 1
r
”−Φ“a− 1
r
”, r →∞, where a ∈ [0,+∞].
The representation ba(r) in terms of the weighted integrals of random fields
ba(r) = D
»ZRn
fr,a(|t|)ξ(t)dt–
was considered in [1].We investigated and obtained various properties of function fr,a(|t|) . In par-
ticular we have proved a representation for it in the form:
fr,a(|t|) =
8<:`
2ar3
´n2P∞m=0 dm(n, r, a, |t|), |t| < r,“
2ar|t|2
”n2
Γ`n2
´P∞m=0 sm(n, r, a, |t|), |t| > r,
(1)
where explicit formulae dm(n, r, a, |t|) and sm(n, r, a, |t|) were derived.For numerical calculations we have obtained speed of convergence in (1), and
the number of terms in this series remainder to achieve given accuracy. Numerousnumerical examples were considered.
References
1. A.Olenko. Tauberian theorems for the fields with OR spectrum. II //The-ory of Prob. 74 (2006).
National Taras Shevchenko University, Ukrainee-mail: [email protected]; [email protected]
162
On the transition density estimates for somediffusion process on d-sets
Victoria Knopova, Ukraine
Let (Γ, ρ,m) be a metric measure space, where Γ ⊂ Rn is compact, ρ is ageodesic metric on Γ , m is a Radon measure with supp m = Γ , and there existc1 , c2 > 0 , such that for a ball B(x, r) , centered at x with radius r , we have
c1rd ≤ m(B(x, r)) ≤ c2r
d, 0 < r ≤ 1.
Let h : R+ → R+ be some increasing function, h(0) = 0 . Consider the approxi-mating Dirichlet form (Er, D(Er))
Er(u) =1
h(r)
ZΓ
ZB(x,r)
|u(x)− u(y)|2
m(B(x, r))m(dy)(m(dx).
Under certain conditions it was shown in [2] that some sequence (rn)n≥0 , rn → 0as n→ 0 , the Γ -limit of (Er, D(Er)) is a local regular Dirichlet form. We wouldlike to employ the structure of this Dirichlet form to show, that the correspondingdiffusion process is the fractional diffusion with parameters, depending on h andd , i.e. that the transition density exists and satisfies certain upper and lowerbounds.
References
[1] M.Barlow, Diffusion on fractals, Lecture notes in Mathematics, Vol.1690,Springer, Berlin (1998).
[2] T.Kumagai, K.T. Sturm, Construction of diffusion processes on fractals, d -sets, and general metric measure spaces, J.Math.Kyoto Univ., 45, N2, 307-327(2005).
[3] K.T.Sturm, Diffusion processes and heat kernels on metric spaces, Ann.Probab., 26, 1-55 (1998).
V.M.Glushkov Institute of Cybernetics, Kiev, Ukrainee-mail: vic− [email protected]
163
Integral equations with stochastic line integrals
A.M. Kolodii, Russia
The main concern of the talk will be related to existence theorems and limittheorems for the equation
ξ(t) = η(t) +
tZ0
S(t, s, ξs, ds), t > 0; (1)
containing a stochastic line integral in a semimartingale field S . For the definition
and investigation of properties of the stochastic line integraltR0
S(u, s, ξs, ds) along
random curve ξ we use some ideas and methods from [1], [2] and [3].Sufficient conditions for existence and uniqueness of a strong continuous solu-
tion and for existence of a weak continuous solution of the equation (1) are givenin [2]. We continue the investigation of the equation (1). The following results areobtained.
(i) Sufficient conditions for existence of a modification with trajectoriesin D (right-continuous with left-hand limits) of the stochastic line integraltR0
S(t, s, ξs, ds) and inequalities for its moments.
(ii) Sufficient conditions for existence of a strong solution of the equation (1)with Borel trajectories.
(iii) Sufficient conditions for existence and uniqueness of a strong solution andfor existence of a weak solution of the equation (1) with trajectories in D .
(iv) Limit theorems for the equation (1).
References
1. Gikhman I.I., Skorokhod A.V. Stochastic Differential Equations and TheirApplications. Naukova Dumka, Kiev, 1982.
2. Kolodii A.M. On conditions for existence of solutions of integral equationswith stochastic line integrals. — Probability Theory and Mathematical Statistics.Proceedings of the 6th Vilnius Conference./ Ed. by B. Grigelionis et al., 1994,405-422.
3. Kolodii A. M. On existence of continuous modifications of random processes.– Theory of Stochastic Processes, 6(22), 2000, no. 1–2, 54-57.
Volgograd State University100 University ave., 400062, Volgograd, Russiae-mail: [email protected]
164
Volterra equations driven by strong martingalekernels
N.A. Kolodii, Russia
Let (Ω,F , (Fz, z∈R2+),P) be a filtered probability space satisfying the usual
assumption (including Cairoli and Walsh condition), T and P denote the σ -algebras of progressively measurable and predictable subsets of R2
+ × Ω respec-tively [1]. Let λ be a locally finite measure on (R2
+,L(R2+)) , where L(R2
+)is the completion of Borel σ -algebra B(R2
+) with respect to the measure λ .Xλ denotes the space of L(R2
+)|B(R) -measurable functions g : R2+ 7→ R with
|||g|||z =“ R
[0,z[
g2dλ”1/2
<∞ for every z ∈ R2+ . We endow the space Xλ with met-
rics %λ(g, g′) =∞Pk=1
2−kh1 ∧ |||g − g′|||(k,k)
i. For g ∈ Xλ and x ∈ R2
+ , let us define
gx ∈ Xλ by gx(u) = g(u)I(u)
[0,x[ . Let Ξλ be the space of progressively measurablerandom fields ξ = (ξ(z), z ∈ R2
+) such that ξ(·, ω) ∈ Xλ for every ω ∈ Ω .The talk is concerned by existence and uniqueness theorems for the stochastic
integral equation
ξ(z) = η(z) +
Z[0,z]
a(z, x, ξz)A(z, dx) +
Z[0,z]
b(z, x, ξx)M(z, dx), z ∈ R2+, (1)
where η ∈ Ξλ , a(z, ω;x, g) ∈ T ⊗ B(R2+) ⊗ B(Xλ)|B(R) ,
b(z;x, ω; g) ∈ B(R2+) ⊗ P ⊗ B(Xλ)|B(R) , M(z;x, ω) ∈ B(R2
+) ⊗ T |B(R) ,A(z;x, ω) ∈ B(R2
+) ⊗ T |B(R) , For every z ∈ R2+ , the random field
(M(z, x), x ∈ R2+) is a strong square integrable martingale, (A(z, x), x ∈ R2
+) isa locally bounded variation field, A(0, x) = M(0, x) = 0 for every x ∈ R2
+ .
Lemma 1) There exist M,A ∈ B(R+) ⊗ T |B(R+) and cM , bA∈ B(R2
+)⊗B(R2+)⊗T |B(R+) such that M(z, x) is a square variation of martingale
M(z, ·) , A(z, x) is a variation of field A(z, ·) , cM(z, y, x) is a square variation ofM(z, ·)−M(y, ·) , bA(z, y, x) is a variation of field A(z, ·)−A(y, ·) on the rectangle[0, x] .
2) If ξ ∈ Ξλ then (x, ω) 7→ ξx(·, ω) ∈ P|B(Xλ) ,
(z, ω;x) 7→ a(z, ω, x, ξz(·, ω)) ∈ T ⊗B(R2+)|B(R) ,
(z;x, ω) 7→ b(z, x, ω, ξx(·, ω)) ∈ B(R2+)⊗P|B(R) .
The definitions, properties and inequalities for moments of stochastic integralswith respect to strong martingale kernels was published in [2] and [3].
Theorem 1 Suppose that there exists a family of measurable non-negative ran-dom fields (Bk(z, x))k=0,1,... such that
165
1) |a(z, x, g)| 6 B0(z, x)(1 + |||g|||z); |b(z, x, g)| 6 B0(z, x)(1 + |||g|||x);
2) if |||g|||z ∨ |||g′|||z 6 k, then |a(z, x, g)− a(z, x, g′)| 6 Bk(z, x)|||g − g′|||z;
3) if |||g|||x ∨ |||g′|||x 6 k, then |b(z, x, g)− b(z, x, g′)| 6 Bk(z, x)|||g − g′|||x;
4)R
[0,z]
Bn(z, x)A(z, dx) +R
[0,z]
B2n(z, x)M(z, dx) 6 Γn(z),
where (Γn(z), z ∈ R2+) — non-random functions, 0 6 Γn(z) 6 Γn(z′) for z 6 z′ .
Then the equation (1) has a solution ξ ∈ Ξλ and this solution is unique in astochastic equivalence sense.Below an increment of the function g : R2
+ 7→ R on the rectangle ]x, y] , x, y ∈ R2+ ,
x < y , is defined by the equality g(]x, y]) = g(y) − g(x1, y2) − g(y1, x2) + g(x) .D denotes the space of right continuous functions which have finite left-hand,left-right-hand and right-left-hand limits.
Theorem 2 Suppose that the conditions of Theorem 1 hold, trajectories of thefield η belong to D and there exist number p > 2 , increasing non-negative func-tions (ϕ(t))t>0 and %(t))t>0 , measurable non-negative random fields B(z, x) andΛ(z, x) , such that
1R0
ϕ(s)s−1−2/pds <∞; E“ zR
0
B2(z, x)Λ(z, dx)”p/2
<∞; limt↓0
%(t) = 0;
0 6 B(y, x) 6 B(z, x), for y < z;
0 6 Λ(y, ]u, v]) 6 Λ(z, ]u, v]) for u < v, y < z;bA(z, y, ]u, v]) 6 Λ(z ∨ y, ]u, v])%(|z − y|);cM(z, y, ]u, v]) 6 Λ(z ∨ y, ]u, v])ϕ2(|z − y|);
|a(z, x, g)− a(y, x, g)| 6 B(z ∨ y, x)%(|z − y|)(1 + |||g|||z);
|b(z, x, g)− b(y, x, g)| 6 B(z ∨ y, x)ϕ2(|z − y|)(1 + |||g|||x).
Then the equation (1) has an unique solution with trajectories in D .
References
1. Gushchin A. A. On the general theory of random fields on the plane. – RussianMath. Surveys. 37 (1982), no. 6(228), 53-74.2. Kolodii N.A. The integrals with respect to a twoparameter strong martin-gale integral kernels end their applications. – Surveys in Applied and IndustrialMathematics, 2004, Vol. 11, No. 1, pp. 120-121.3. Kolodii N.A. The inequalities for integrals with respect to a continuous strongmartingale. – Vestnik VolSU. 1. Mathematics. Physics. Vol. 8, 2003-2004, 35-47.
Volgograd State University100 University ave., 400062, Volgograd, Russiae-mail: [email protected]
166
Fractional stability of diffusion approximationfor random differential equations
Yuriy V. Kolomiets, USA
We consider the systems of random differential equations. The coefficients ofthe equations depend on a small parameter. The first equation, "slow" component,ordinary differential equation, has unbounded highly oscillating in space variablecoefficients and random disturbances, which are described by the second equa-tion, "fast" component, with periodic coefficients. The sufficient conditions forweak convergence as small parameter goes to zero of the solutions of the "slow"components to the certain random process are proved. Classical Diffusion Approx-imation Theorem (DAT) says, that drift coefficient of the approximated StochasticDifferential Equation (SDE) include a derivative with respect to a space variable ofthe unbounded coefficients (see, e.g. monograph of A. Skorokhod [6], and bibliog-raphy). So, we can not apply classical DAT because of highly oscillating characterof dependency on the small parameter of the unbounded coefficient of the "slow"component. From the other hand we can not to apply the Limit Theorem for SDE(in the sense of G. Kulinich [2], N. Portenko [5], M. Freidlin and A.D. Wentzell [1],S. Makhno [3,4]) because the "slow" component is ODE and consequently has nononzero diffusion coefficient (the presence of strongly positive diffusion coefficientis a necessary conditions for such kind of the theorems).
Refs: [1] M. Freidlin, and A. D. Wentzell. Necessary and sufficient conditionsfor weak convergence of one-dimensional Markov processes. In Festschrift Ded-icated to 70th Birthday of Professor E. B. Dynkin (M. Freidlin, ed.), 1994,pp.95–109. Birkhauser, Boston
[2] G.L. Kulinich. On necessary and sufficient conditions for the convergence ofsolutions of one dimensional diffusion stochastic equations with a non-regular de-pendence of coefficients on a parameter. Theory Prob. Appl., 27 (1982), 4,pp.795–802
[3] S.Ya. Makhno. Stochastic equations with unbounded drift. Theory of Stoch.Proc. V. 4(20)(1998), 1–2, pp. 211–221
[4] S.Ya. Makhno.The convergence of solutions one-dimensional stochasticequations.Theory Prob. Appl.,44 (1999), 3 pp. 555–572,(Russian)
[5] N.I. Portenko.To the theory of the generalized diffusion. Lecture Notes inControl and Information Sciences, (1986), 78
[6] A.V. Skorokhod. Asymptotic methods of the theory stochastic differentialequations. Naukova Dumka, Kiev, 1987 (Russian)
Kent State University; IAMM NAS of Ukraine,PO Box 5190, Kent, OH 44242-0001 100111Department of Mathematical Sciencese-mail: [email protected]
167
Integral Representation of Feller Semigroupsthat Describes Diffusion process on a Half-Line
with Non-Local Boundary Conditions
Kononchuk P.P., Ukraine
Let D = x ∈ R1 : x > 0 , S = 0 - the boundary of D andD = x ∈ R1 : x ≥ 0 is the closure of D . Let us assume that on D is given agenerating second-order differential operator in the space of all two times differ-entiable operators with compact carrier C2
K(D) :
Lϕ(x) =1
2b(x)
d2ϕ
dx2+ a(x)
dϕ
dx, (1)
where a(x) and b(x) are bounded continuous functions on D and b(x) ≥ 0 . Letus consider that in the point x = 0 is given an integral operator i.e. mapping fromC2k(D) into the space of all continuous functions on boundary S :
L0ϕ(0) =
ZD
“ϕ(0)− ϕ(y)
”π(dy), (2)
where π is a nonnegative Borel measure on D , 0 < π(D) < +∞ . Notice (see [1])that right-hand member of (2) is an addend in general boundary conditions forone-dimension diffusion processes that corresponds to the possible jumps of theprocess from its boundary S into D .
We define the next problem: a construction of a operator semigroup Tt , t ≥ 0 ,that describes a Feller process on D such that on D it coincides with a diffusionprocess handled by an operator L and its behavior on a boundary S is definedby a boundary condition L0ϕ(0) = 0 .
In order to solve the problem we will use analytical methods. It means that thesemigroup can be found as a solution of a corresponding parabolic boundary prob-lem with non-local boundary conditions. Classical solution for it will be obtainedby a method of boundary integral equations with some additional restrictions tothe L from (1) and π from (2) (see [2]).
References
1. W. Feller. the parabolic differential equations and the associated semigroups oftransformation // Ann. Math. 1952, v. 55, #3, p. 468-519.
2. O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’tseva. Linear and Quasi-linear Equations of Paraboloc Type – Nauka, Moscow, 1967, 736 p.
Lviv, 79001, Universytetska, 1, LNU after Ivan Frankoe-mail: [email protected]
168
An analytical approach to the two-fluid theoryof Herman and Prigogine for town traffic
D. G. Konstantinides, Greece; N.U. Prabhu, USA
In a pioneering study Herman and Prigogine [2] developed a two-fluid modelfor town traffic, the two fluids consisting of moving cars and cars stopped as aresult of congestion, traffic signals, stop signs or other traffic conditions. The mainquantities of interest are the stop time per unit distance and the trip time perunit distance. Their theory was validated empirically in several cities, including theDetroit metropolitan area, London (U.K.), Melbourne (Australia) and Brussels.They found that these quantities are linearly dependent (see also [1]).
In this paper we use an analytical approach to verify the findings of theseauthors. Specifically, we consider a fixed-cycle traffic light and cars moving intwo opposing lanes (say, west to east and south to north). The phenomenon ofcongestion occurs in a sequence of cycles, each cycle consisting of a red phase oflength r and a green phase of length g , where r and g are fixed numbers andc = r+g is the cycle time. We denote by Zi(t) the number of cars in lane i ( i = 1is the east bound lane and i = 2 is the north bound lane). During a red phase Z1(t)is the number of cars waiting to cross the intersection (stopped cars), and Z2(t) isthe number of cars in the process of crossing (moving cars). During a green phasethe reverse statement holds. We propose a deterministic linear model in which thearrival of new cars at the traffic light and departures from the intersection occurat given constant rates in each lane. Our main objective is to study the nature oflong run dependence between Z1(t) and Z2(t) .
We propose an analytical model for vehicular traffic at a fixed-cycle trafficlight. It is found that in the long run the numbers of vehicles in the two opposinglanes are linearly dependent, confirming the empirical findings of Herman andPrigogine.
References
[1] De Palma (2003) In Memoriam Ilya Prigogine (1917-2003). TransportationScience. 37, No. 3, 255–256.
[2] Herman, R. and Prigogine, I. (1979) A two-fluid approach to town traffic.Science. 204, 148-151.
Department of Statistics and Actuarial - Financial Mathematics,University of the Aegean, Karlovassi, GR-83 200 Samos, Greecee-mail: [email protected]
Department of Operations Research and Industrial Engineering,276 Rhodes Hall, Cornell University, Ithaca, New York 14853-3801, USAe-mail: [email protected]
169
On the absolutely continuous spectrum ofblock operator matrices
Oleksii Konstantinov, Ukraine
In this work we extend recent results by S.Albeverio, K.Makarov, A.Motovilov[2] and S.Denisov [3] on the preservation of the absolutely continuous spectrumfor block operator matrices. Consider a Hilbert space H = H1 ×H2, which is theorthogonal sum of two Hilbert spaces H1 and H2, and a self-adjoint operator Hin H given by a block operator matrix
H = A+ V =
A1 0
0 A2
!+
0 B
B∗ 0
!=
A1 B
B∗ A2
!.
We denote by S2(H,K) the class of Hilbert-Schmidt operators from H into K .EH , σess(H), and σac(H) denote the spectral measure, essential spectrum, andabsolutely continuous spectrum of a self-adjoint operator H, respectively. Hac
denotes the absolutely continuous part of the operator H . The part of H actingin EH(δ)H is denoted by H(δ).
Theorem 1 Let A1 , A2 be self-adjoint operators in Hilbert spaces H1 , H2 re-spectively. Assume that B ∈ S2(H2,H1) and set ∆ := R \ (σess(A1) ∩ σess(A2)).Then the local wave operators W±
∆ (H,A) exist and are complete. In particularHac(∆) and Aac(∆) are unitarily equivalent and
σac(H) ∩∆ = (σac(A1) ∪ σac(A2)) ∩∆.
Corollary 2 Suppose in addition that the set σess(A1) ∩ σess(A2) has Lebesguemeasure zero. Then the wave operators W±(H,A) exist and are complete. Inparticular the operators Hac and Aac are unitarily equivalent.
The talk is based on the joint work with S.Albeverio [1].
References
[1] Albeverio S., Konstantinov A., On the absolutely continuous spectrum ofblock operator matrices University of Bonn, SFB 611, Preprint no. 243(2005).
[2] S. Albeverio, K. Makarov, A. Motovilov, Graph subspaces and the spectralshift function, Canad. J. Math., 55 (2003), 449–503.
[3] S.A. Denisov, On the preservation of absolutely continuous spectrum forSchrodinger operators, J. Funct. Anal. 231 (2006), 143–156.
Department of Mathematical AnalysisUniversity of Kyiv64 Volodymyrska str., 01033 Kyiv, Ukrainee-mail: [email protected]
170
Linear regression with trending variables andconstraints on parameters
Arnold Korkhin, Ukraine
Let’s consider regression yt = xᵀt α
0 + εt, t = 1, T , where yt ∈ <1, xt ∈ <n,α0 ∈ <n, εt ∈ <1 is noise, the character ” > ” means transposition.
Let’s estimate α0 , having solved a problem
1
2T
TXt=1
(yt − xᵀt α)2 ⇒ min, gi(α) = aᵀ
i α−Bi ≤ 0, i ∈ I, (1)
where xt, yt, t = 1, T and ai ∈ <n, Bi ∈ <1, i ∈ I are known values.The constraints in (1) are determined from practical reasons. The true value
of parameters of regression α0 can satisfy constraints:
gi(α0) = 0, i ∈ I0
1 ; gi(α0) < 0, i ∈ I0
2 ; I01∪I0
2 = I = 1, . . . ,m, I01∩I0
2 = , m ≤ n.(2)
Let’s consider following for independent variable, constraints and noise.Assumption 1. The random variables εt are centered and are independent. Theydo not depend from xt, t = 1, 2, ... , have variance σ2 and distribution functionsΦt (u) , t = 1, 2, ... , while supt=1,2,...
R‖u‖>cu
2dΦt (u) → 0 , at c→∞ .
Let’s designate: ZT =TPt=1
xtxᵀt , ET = diag
“pzT11,
pzT22, ...,
pzTnn
”, where
zTjj , j = 1, n is an element on a main diagonal ZT ; A is a matrix with rows thatare rows of aᵀ
i , i ∈ I ; B ∈ <m is right part of constraints in (1).Assumption 2. The matrix ZT is not singular for all T . The matrix A has afull rank.Assumption 3. The matrix RT = E−1
T ZTE−1T → R at T → ∞ , where R is a
positive definite matrix.
Assumption 4. zTjj →∞,x2
j,T+1
zTjj
→ 0, j = 1, n at T →∞ .
Assumption 5. There exists such diagonal matrix ET (m×m) with positiveelements on a main diagonal EiT , i ∈ I for which there is a limit of a matrixGT = ETAE
−1T : lim
T→∞GT = G .
The assumption 5, in particular, is satisfy, when the independent variable haveconstant mean. Then ET = ET =
√TJn, where Jn is a unit matrix (n× n) .
Assumption 6. The matrix G1 (m1 × n) has a full rank, the matrix G1
(m1 × n) consists out of rows of a matrix G with indexes i ∈ I01 (m1 is a number
of elements in set I01 ).
It is supposed that independent variable xt, t = 1, 2, ... is nonrandom. If
there is a intercept, then in the problem of estimation (1) αᵀ =ˆα1
...αᵀ˜ᵀ , where
α1 ∈ <1, α ∈ <n−1 , xᵀ =ˆ1...xᵀ˜ᵀ , xᵀ ∈ <n−1 . Often there is no constraint
on the intercept misses. In this case the solution of a problem of estimation can
171
be simplified by reducing quantity of variables by 1. Let’s designate Om zerom -dimensional vector.
Theorem 1. If an assumption 2 is true, A =ˆOm
...A˜. Then estimation of para-
meter of regression looks like αᵀT =
ˆαT1
...αᵀT
˜ᵀ , where αT1 = ym − αᵀT xm, αT
is the solution of the problem
1
2αᵀzTα− αᵀd⇒ min, Aα ≤ B. (3)
Here zT =TPt=1
(xt − xm) (xt − xm)ᵀ , d =TPt=1
(xt − xm) (yt − ym) , B ∈ <m is a
vector of right parts of constraints in (1), xm =TPt=1
xt/T, ym =TPt=1
yt/T .
Based on the theorem, the estimation may be reduced to the solution of aproblem of quadratic programming. The matrix of this problem has elements withabsolute value not greater than 1, which increases the accuracy of estimation.Theorem 2. Assumptions 1-6 be true, then the solution of a problem (1) αT isa consistent estimation α0 .Theorem 3. If the assumptions 1-6 are true, then random variableUT = ET
`αT − α0
´at T → ∞ converges in distribution to a random variable
U, which is solution of a problem:
1
2XᵀRX −QᵀX ⇒ min, G1X ≤ Om1 , (4)
where Q ∼ N(On, σ2Jn).
The estimation of mean square error matrix of estimations of parameters ofregression is offered in the report KT = σ2
T kT , where kT = MuTu>T | αT , σT ,
σ2T =
`T − n+
Pi∈IγiT
´−1TPt=1
(yt − xᵀt αT )2 .
The sense of introduced designations is following. A random variable uT ∈ <nis solution of a problem
1
2zᵀRT z − qᵀ
T z ⇒ min, GT z ≤ BT , (5)
where z ∈ <n , qT ∼ N(On, σ2TJn). Then components BT are determined by
expression biT =ET (Bi−a
ᵀi αT )
σT(1− γiT ) , i ∈ I .
Value γiT = 1, if |gi (αT )| ≤ ξ; γiT = 0, if |gi (αT )| > ξ, where ξ is number,for which one −ξ > gi
`α0´, i ∈ I0
2 . It is proposed to take the accuracy ofcalculations on the computer as ξ .
The consistency of an estimator KT is proved: p limT→∞KT = M˘UU>
¯.
National Mining University, Department of Economic Cybernetics & IT,19 Karl Marx Avenue, Dnepropetrovsk, 49600 Ukrainee-mail: [email protected]
172
On wavelet expansion of some randomprocesses
Yu. V. Kozachenko, M.M. Perestyuk and O.I. Vasylyk, Ukraine
We present conditions for uniform convergence with probability one of waveletexpansion of ϕ -sub-Gaussian (in particular, Gaussian) random processes definedon the space R .
In some papers concerning wavelet expansions of random processes it is as-sumed that sample paths of all stationary processes are bounded on R with prob-ability one, but in general this is not so. The majority of random processes, whichare interesting from the theoretical point of view as well as from possible ap-plications, have almost surely unbounded sample paths on R . Among them, forinstance, there are Wiener process, the processes of fractional Brownian motion,some classes of stationary processes, etc.
We derive general theorem on uniform convergence of the wavelet expansionson finite interval for the functions, which have some order of growth on infinity.We consider a system of wavelets, for which f -wavelets φ(x) and m -waveletsψ(x) satisfy conditions
|φ(x)| ≤ Φ(x), |ψ(x)| < Φ(x),
where Φ(x) is some even bounded monotonically decreasing as x > 0 function.This theorem is applied to studying conditions for uniform convergence with
probability one of the wavelet expansion of ϕ -sub-Gaussian random processes. Inparticular, it is shown that for such systems of wavelets that
Φ (x) =1
1 + |x|b, b > α+ 1,
wavelet expansions of the processes of fractional Brownian motion Zα(t) withHurst index α converge uniformly with probability one on any finite interval.
References
1. Yu. V. Kozachenko. Lectures on wavelet analysis. (Ukrainian), TViMC,Kyiv (2004).
2. V. V. Buldygin, Yu. V. Kozachenko. Metric characterization of randomvariables and random processes. American Mathematical Society, Providence RI(2000).
3. Yu. Kozachenko, T. Sottinen, O. I. Vasilik. Self-similar processes withstationary increments in the spaces Subϕ (Ω) . Theory of Probability and Math.Statistics, No. 65, 77 – 88 (2002).
Mechanics and Mathematics Faculty,Kyiv National University, Kyiv 01033, Ukrainee-mail: [email protected], [email protected], [email protected]
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On one method of simulation of stochasticprocesses
Yuriy Kozachenko, Iryna Rozora, and Yevgeniy Turchyn, Ukraine
Let (Ω,F , P ) be a probability space and T be a parametric set. Consider acentered second-order stochastic process X(t), t ∈ T, with its correlation functionR(t, s) = EX(t)X(s) .
Let (Λ,BΛ, µ) be a measurable space with σ -additive measure µ .The following theorem holds true.
Theorem 1 Let f(t, λ), t ∈ T, belong to L2(Λ, µ) and a family of functionsgk(λ), k ∈ Z , be an orthonormal basis in L2(Λ, µ) .
Then the correlation function R(t, s) has the representation as
R(t, s) =
ZΛ
f(t, λ)f(s, λ)dµ(λ),
if and only if the process X(t) is represented as
X(t) =Xk∈Z
ak(t)ξk, (1)
whereak(t) =
ZΛ
f(t, λ)gk(λ)dµ(λ),
ξk are centered non-correlated random variables, Eξk = 0, Eξkξl = δkl andEξ2K = 1 .
If in theorem 1 the process X(t) is Gaussian then ξk, k ∈ Z , are independentGaussian random variables.
Consider an example which deals with wavelet basis.
Example 1 Suggest that the correlation function can be represented as
R(t, s) =
Z ∞
−∞f(t, λ)f(s, λ)dλ.
Consider wavelet basis ϕ0k(λ), ψjk(λ), j = 0,∞, k ∈ Z .It’s well-knownfact that this system is orthonormal basis in the space L2(R)(see, for example, in[2]). Then
X(t) =Xk∈E
ξ0kα0k(t) +
∞Xj=0
Xk∈E
ηjkβjk(t),
whereα0k(t) =
ZRf(t, λ)ϕ0kdλ, βjk(t) =
ZRf(t, λ)ψjkdλ.
If the process X(t) is stationary with spectral density function f(λ) thenf(t, λ) = eitλ
pf(λ).
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The expansion (1) can be used for model construction of stochastic processes indifferent functional spaces with given accuracy and reliability.
Let X(t) =∞Pk=0
zk(t)ξk be Gaussian stochastic process; ξk, k = 0, 1, 2 . . . are
independent centered Gaussian random variables, Eξk = 0, Eξ2k = 1 .
Definition 1 A process XN (t) is called a model of the process X(t) if
XN (t) =
NXk=0
zk(t)ξk.
Let || · || be a norm in some functional space, for example, in Lp([0, T ])thenorm will be ||f(t)|| = (
R T0|f(t)|pdt)
1p .
Definition 2 The model XN (t) approximates stochastic process X(t) with givenaccuracy δ > 0 and reliability 1− ν, ν ∈ (0, 1) if
P||XN (t)−X(t)|| > δ ≤ ν. (2)
To construct the model with given accuracy δ and reliability 1 − ν in givenBanach space it’s necessary to find a such N for which the inequality (2) is fulfilled.The methods for finding N in different Banach spaces with given accuracy andreliability are investigated in [1], [3] and others.
References
[1] Yu.V. Kozachenko, A.O. Pashko Simulation of stochastic processes, “ KyivUniversity”, 1999. (in Ukrainian)
[2] I. Daubechies. Ten lectures on Wavelets. Society for industrial and appliedmathematics. Philadelphia (1992).
[3] Yu. V. Kozachenko and I. V. Rozora, Reliability and accuracy of stohasticprocesses from the space Subϕ(Ω) . Probab. Theory and Math. Statist., no.71,2004, pp. 93-105
Kyiv National Taras Shevchenko University,Volodymyrska str., 64, Kyiv, 01033e-mail: [email protected] National Taras Shevchenko University,Volodymyrska str., 64, Kyiv, 01033e-mail: [email protected] State Agricultural University,Voroshylova str., 25, Dnipropetrovske-mail: [email protected]
175
On some conditions of Gaussian generalizedhomogeneous fields distribution equivalence
Krasnitsky Serge, Ukraine
If Gaussian homogeneous fields have the spectral densities of power type, thenthe conditions of their probabilistic measures equivalence may be formulated as theexistence of their mean values ∆a and correlation functions ∆r difference certainsmoothing. It is noticed [1] that fractional differentiation in the sense of Lizorcin[2] is suitable for such conditions formulation. But the checking of this conditionsfulfilling is often complicated by the circumstance that in according to definitionsthe above mentioned conditions are to be formulated for some ∆a , ∆r extensions,but not for ∆a , ∆r directly. At the same time the results for some partial casesof power spectrum, for example, for the spectrum of isotropic type, are free of thisuncomfortable circumstance [3]. In the given report the conditions of equivalenceare formulated directly in the terms of functions (a, (r for the spectral densitiesof rather wide no isotropic class and are generalized some results of [1, 3].
To escape the cumbersome expressions we’ll look only the case of two randomfields with different mean values, and what’s more, the first field has the zeromean value and the second — the mean value a(ϕ) , ϕ ∈ Φ = C∞0 (T ) (T is opensubset of RN ). In the given theses we also shall not give the result formulationfor maximal universal type of spectrum as we can, but only for one concrete case.By assumption, every looked field has the spectral density f(λ) , λ ∈ RN , that issatisfied the inequality c1u(λ) ≤ f(λ) ≤ c2u(λ) , where
u(λ) =
1 +
NXj=1
|λj |2lj!
1 +
NXj=1
|λj |2mj
!−1
,
l1, . . . , lN ,m1, . . . ,mN are nonnegative. Let T = (α1, β1) × . . . × (αn, βn) is arectangle with sides, parallel to coordinate axes in RN . For generalized functiong , basic function ϕ ∈ Φ , natural number j ≤ N , real number s and h > 0 let’sdefine:
˙δsj , ϕ
¸= 〈g, ϕ(y1, . . . , yj − s, . . . , yN )〉 , ∆s
j = δsjg − g ,
∆∂s,hj g =
( R h−h
DD
[s]j ∆s
jg, ϕE|x|−(1+s)dx for s 6= 0,˙
Dsjg, ϕ
¸for s = 0.
where Dkj = ∂k/∂ykj , [s] — is the entire part s , s — the fractional part. Let T j,h
be the rectangle that is the same as T if the number lj is entire and in anothercase is obtained of T by changing the interval (αj , βj) for interval (αj−h, βj+h)with h > 0 . Let’s also Φj,h = C∞0 (T j,h) .
Theorem 1 For the equivalence of measures corresponding to the looked fields it’snecessary (sufficient) that for every (some) sufficiently small h > 0 the functiona(ϕ) could be representative in form a(ϕ) =
PNj=1 ∆∂
lj ,h
j cj(ϕ) , ϕ ∈ Φ , wherethe generalized functions cj are defined on Φj,h and are those that for every
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i ∈ 1, . . . , N and for almost every y ∈ (−h, h) functional D[mi]i ∆y
j cj (Dmij cj
for entire mi ) on Φ is the function by ∈ L2(T ) (b for entire mi ) and moreover,the inequalityR(−h,h)
‖by‖2L2(T )|y|−(1+2mi)dy < +∞ is take place for every i for which thenumber mi is not entire.
References
[1] Krasnitsky S.M. On conditions of equivalence of Gaussian measures corre-sponding to homogeneous random fields with different mathematical expecta-tions //Dopov. Nats. Akad. Nauk Ukr. - 1998. - 2. - С.35-39.
[2] Krasnitsky S.M. Generalized Liuville differentiation and functional spacesLrp(En) //Izvestiya AN SSSR. Ser. Mat. - 1965. - 29, 1. - С.109-126.
[3] Yadrenko M.I. On absolute continuity of measures corresponding to Gaussianhomogeneous random fields //Theory Probab. Math. Stat. - 1972. - 7. - С.152-161
Kyiv National University of Technology and Designe-mail: [email protected]
A limit theorem for stochastic equation withlocal time
Ivan Krykun, Ukraine
We consider stochastic equation with local time:
ξ(t) = x+ βLξ(t, 0) +
Z t
0
b(ξ(s))ds+
Z t
0
σ(ξ(s))dw(s).
In this equation Lξ(t, 0) is the symmetric local time of the process ξ(t) at
the point 0, |β| < 1 , functions b(x), σ(x),
xZ0
b(y)
σ2(y)dy is uniformly bounded.
We investigate limit behavior ofξ(t)
tαif t→∞ for different α.
For example, if α = 12
then the limit process is a solution of such stochasticequation :
ξ(t) = γLξ(t, 0) +
Z t
0
A(ξ(s))dw(s).
Institute of Applied Mathematics and Mechanicsof the National Academy of Sciences of Ukraine,R.Luxemburg str., 74, Donetsk, Ukraine, 83114e-mail: [email protected]
177
Reselling of European optionAlexander Kukush, Yuliya Mishura, Georgiy Shevchenko, Ukraine
On Black and Scholes market investor buys a European call op- tion. In clas-sical setting, Investor is not entitled to exercise the option before the maturityand should wait until the maturity. However, it is known that on real financialmarkets he has an opportunity to resell the option before the maturity. Thus, aninvestigation of the reselling problem is essential, while, to the authors’ knowledge,there is no paper dealing with this problem.
We treat the following model. On the Black–Scholes security market with aninterest rate r , at the moment t0 = 0 , Investor buys a European call option withthe strike price K and the maturity T , on the stock with initial value S0 , for theprice CBS(S0, T ) = CBS(S0, T ;σ,K; r) computed by the Black–Scholes formula.At any moment t ∈ (0, T ) he can resell the option for a certain market price Cmt ,which may differ from the “fair” price CBS(S0, T − t) . We propose a stochasticmodel of the market price Cmt , which does not lead to an arbitrage opportunity.It is shown that such an option, with reselling possibility, is equivalent to certainAmerican type derivative. This allows to describe the optimal reselling time forthe option in terms of nonrandom stopping sets Gt , which are subsets of thetwo-dimensional phase space R+ × R+ 3 (St, C
mt ) . It is natural to establish the
threshold structure for the stopping sets in the problem of reselling of a Euro-pean option, in particular, that Gt is a set of points lying above a certain curve.We establish similar threshold structure for stopping sets in the simplified modelof market price for European option, where stochastic volatility process has nomemory. A numerical algorithm for construction of the optimal stopping set isgiven.Department of Mechanics and Mathematics,Kyiv National Taras Shevchenko University,64 Volodymyrska, 01033 Kyiv, [email protected], [email protected], [email protected]
On non-equilibrium stochastic dynamics forinteracting particle systems in continuum
Oleksandr Kutovyy, Germany
We give an abstract approach for the construction of non-equilibrium sto-chastic dynamics for interacting particle systems in continuum. As example, weconstruct the non-equilibrium Glauber dynamics as a Markov process in config-uration space for an infinite particle system in continuum with a general class ofinitial distributions. This class we define in terms of correlation functions boundsand it is preserved during the Markov evolution we constructed.Faculty of Mathematics, University of Bielefed,Postfach 100131, D-33501 Bielefeld, Germanye-mail: [email protected]
178
Gaussian approximation of cumulative incomeprocess for [G|GI|∞]r -networks
Lebedev E.A., Makushenko I.A., Ukraine
The basic model under consideration in the paper is a multi-channel networksof queues. From the outside calls arrive at the i−th node in the instant of timeτ ik, k = 1, 2, . . . ; νi(t) is the total numbers of calls arrived in the i−th node inthe time interval [0, t], ν′(t) = (ν1(t), . . . , νr(t)). Each of "r" nodes is a multi-channel queuing system. We will denote a distribution function of the service timethrough Gi(t), i = 1, . . . , r. After service in the i -th node the call arrives to thej−th node with the probability pij and leaves the network with the probabilitypir+1 = 1−
Prj=1 pij , P = ‖pij‖r1 is a switching matrix of network. The additional
node numbered "r + 1" is interpreted as "exit" from the network.We will define the service process as an r -dimensional process
X ′(t) = (X1(t), . . . Xr(t)), where Xi(t) is the number of occupied servers inthe i -th node at the instant of time t. At the process X(t) we will set an additivefunctional Y ′(t) = (Y1(t), . . . , Yr(t)), where Yi(t), i = 1, . . . , r is the number ofcalls served in the i -th node during the time interval [0, t]. Since Y (t) defines thetotal income for the network in the interval [0, t] then we will refer to Y (t), t ≥ 0as the cumulative income process.
For the simulation of the service process jointly with the cumulative incomeprocess we will use an 2r -dimensional branching Bellman-Harris process(χm1 (t), . . . , χmr (t), γm1 (t), . . . , γmr (t)) = ((χm(t))′, (γm(t))′), m = 1, . . . , 2r, whichis given by the generating functions of direct offsprings:
f i(x, y) = yi(
rXj=1
pijxj + pir+1), fr+i(x, y) = yi, i = 1, . . . , r.
A lifetime of particles has the following distribution function: Gi(t) for i = 1, . . . , rand Gi(t) = e(t− 1) for i = r + 1, . . . , 2r (e(·) is the function of unit jump). Inconnection with ((χm(t))′, (γm(t))′) we introduce the notations:
χ(t) = ‖χmi (t)‖r1, Γ(t) = ‖γmi (t)‖r1, P (t) = ‖pmi (t)‖r1 = ‖Eχmi (t)‖r1,
A(t) = ‖Ami (t)‖r1 = ‖Eγmi (t)‖r1.
The vector process (X ′(t), Y ′(t)) we will study in heavy traffic regime. Itmeans that parameters of the input flow and the service time depend on "n" (thenumber of series) in such a way that the following conditions hold true.
1) For νn(t) there are constants λi ≥ 0, i = 1, . . . , r, λ1 + · · ·+ λr 6= 0, that
n−1/2(ν(n)1 (t)− λ1nt, . . . , ν
(n)r (t)− λrnt)
U⇒W ′(t) = (W1(t), . . . ,Wr(t)),
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where ν(n)(t) = νn(nt), W (t) is an r -dimensional process of Brownian mo-tion with the null-vector of mean values EW (1) = 0 and the correlation matrixEW (1)W ′(1) = σ2 = ‖σij‖r1.
2) G(n)i (t)
d⇒ Gi(t), as n→∞, i = 1, . . . , r, where G(n)i (t) = Gni (nt).
For random vectors ξ′ = (ξ1, . . . , ξr), η′ = (η1, . . . , ηr) and random matries
Ξ(t) = ‖ξij(t)‖r1, H(t) = ‖ηij(t)‖r1 with components depending on time t ≥ 0 weintroduce the following notations: R(ξ, η) = Eξη′ − EξEη′,
R(Ξ(s), H(t)) = E[Ξ′(s)∆(λ)H(t)− Ξ′(s)∆(λ)H(t)], s < t,
where Ξ(s) = ‖Eξij(s)‖r1, H(t) = ‖Eηij(t)‖r1, ∆(λ) = ‖λiδij‖r1, is fixed nonran-dom diagonal matrix.
To construct an approximate process for (Xn′(t), Y n′(t)), t ≥ 0 it is necessarytwo independent Gaussian processes:
(ξ(1)′(t), η(1)′(t)) d= (
Z t
0
dW ′(u)P (t− u),
Z t
0
dW ′(u)A(t− u))
and (ξ(2)′(t), η(2)′(t)) with the null vector of mean values and the correlationcharacteristics: R(ξ(2)(t), ξ(2)(t)) =
R t0R(χ(u), χ(u))du ,
R(η(2)(t), η(2)(t)) =R t0R(Γ(u),Γ(u))du ,
R(ξ(2)(t), η(2)(t)) =R t0R(χ(u),Γ(u))du and for s < t .
R(ξ(2)(s), ξ(2)(t)) =R s0R(χ(u), χ(u+ t− s))du ,
R(ξ(2)(s), η(2)(t)) =R s0R(χ(u),Γ(u+ t− s))du ,
R(η(2)(s), η(2)(t)) =R s0R(Γ(u),Γ(u+ t− s))du ,
R(η(2)(s), ξ(2)(t)) =R s0R(Γ(u), χ(u+ t− s))du.
Let us consider the sequence of stochastic processes (ξ(n)′(t), η(n)′(t)),n = 1, 2, . . . , where ξ(n)′(t) = n−1/2(Xn′(nt) − nλ′
R t0P (n)(u)du),
η(n)′(t) = n−1/2(Y n′(nt)−nλ′R t0A(n)(u)du, P (n)(t) = Pn(nt), A(n)(t) = An(nt),
matries Pn(t), An(t) are given in the same way as P (t), A(t) with substitutionGni (t) in place of Gi(t).
Theorem. Let at the starting moment the stochastic network of type[G(n)|GI(n)|∞]r be empty, the conditions 1), 2) be satisfied and the spectral radiusof the switching matrix P be strictly less than 1. Then for any finite interval [0, T ]the sequence of stochastic processes (ξ(n)′(t), η(n)′(t)), n ≥ 1 converges weaklyin uniform topology to (ξ(1)′(t) + ξ(2)′(t), η(1)′(t) + η(2)′(t)), as n→∞.
The approximate process can be use for construction of objective functions inincome optimization problems for [G|GI|∞]r -models.
Department of Applied Statistics, Faculty of CyberneticsKiev University, Vladimirskaya str.,64, Kiev, 01033, Ukrainee-mail: [email protected]
180
Student processes
Nikolai Leonenko, UK
It is now generally accepted that heavy-tailed distributions occur commonlyin practice. Their use is now widespread in communication network, risky assetand insurance modelling. However, the study of stationary processes having theseheavy-tailed distributions as their one-dimensional distributions and also havinga full range of possible dependence structures has received rather little attention.In this work we focus on such processes with Student t marginals. The Student tfamily with ν degrees of freedom covers the range of power tail possibilities, thespectrum including the Cauchy distribution (ν = 1) and ranging through to theGaussian as ν →∞ .
In the field of finance, distributions of logarithmic asset returns can oftenbe fitted extremely well by Student t -distribution. In particularly a number ofauthors have advocated using a t -distribution with ν degrees of freedom, typically3 ≤ ν ≤ 5. This implies infinite k th moments for k ≥ ν.
Another issue in modelling economic and financial time series is that theirsample autocorrelation functions (acf) may decay quickly, but their absolute orsquared increments may have acfs with non-negligible values for large lags. Theseubiquitous phenomena call for an effort to develop reasonable models which can beintegrated into the economic and financial theory as well as theories of turbulence.This approach has a long history, certainly dating back to Mandelbrot’s work inthe 1960s, in which use of heavy-tailed distributions (stable or Pareto type) wasadvocated.
We propose a number of stochastic processes with Student marginals andvarious types of dependence structures, allowing for both short- and long-rangedependence. A particular motivation is the modelling of risky asset time seriesrelated to a fractal activity time geometrical Brownian motion (FATGBM). Weare able to prove the existence of a FATGBM model and pricing formulae.
This talk is based on the joint work with C.C. Heyde (Australian NationalUniversity and Columbia University).
References
Heyde, C. C.; Leonenko, N. N. Student processes. Adv. in Appl. Probab. 37(2005), no. 2, 342—365.
Cardiff School of Mathematics, Cardiff University, UKe-mail: [email protected]
181
Uniqueness in Law of Solutions of StochasticDifferential Inclusions
Andrei N. Lepeyev, Belarus
The following stochastic differential inclusion (SDI) is considered
dXt ∈ B(Xt) dWt, t ≥ 0, (1)
where B : IR → comp(IR) - multi-valued Borel measurable mapping,comp(IR) - the set of all non-empty compact subsets of IR with Haus-dorff metric ρ(A,B) = max(β(A,B), β(B,A)),∀A,B ∈ comp(IR) , whereβ(A,B) = supx∈A(infy∈B |x − y|) - excess of A over B , W - one-dimensionalWiener process.
The existence and uniqueness of weak solutions of SDIs were investigated bymany mathematicians. Sufficient existence conditions were given in [1], [2], andother works. Paper [3] introduced necessary and sufficient existence conditionsfor weak solutions of inclusion (1). The objective is to find the uniqueness in lawconditions for solutions of inclusion (1).
Stochastic process (X, IF) , defined on probability space (Ω,F ,P) with filtra-tion IF = (Ft)t≥0 , is called a weak solution of SDI (1), if there exist Wienerprocess (W, IF) with W 0 = 0 and F -measurable process (u, IF) , such thatu(t, ω) ∈ B(X(t, ω)) l+ ×P -a.e. and the following holds P - a.s.∀t ≥ 0
Xt = X0 +
Z t
0
u(s) dW s.
Definition 1 Stochastic process (X, IF) , defined on probability space (Ω,F ,P)with filtration IF = (Ft)t≥0 , is called an explicit weak solution of SDI (1), ifthere exist Wiener process (W, IF) with W 0 = 0 and Borel measurable functionv : IR → IR , such that v(x) ∈ B(x),∀x ∈ IR and the following holds P - a.s.∀t ≥ 0
Xt = X0 +
Z t
0
v(Xs) dW s.
We say that the uniqueness in law holds for SDI (1) if any two solutions(X1, IF1) and (X2, IF2) with coinciding initial distributions possess the same im-age law on the space of continuous functions over IR .
The uniqueness in law can be considered in the class of all explicit solutions.Definition 2 We say that the uniqueness in law with respect to selection v holdsfor SDI (1) if any two solutions (X1, IF1) and (X2, IF2) with respect to selection vwith coinciding initial distributions possess the same image law on the space ofcontinuous functions over IR .
182
Definition 3 We say that the selectiowise uniqueness in law holds for SDI (1) ifuniqueness in law holds for every explicit Borel measurable selection of the SDIright side.
We need the following sets to state the main results
MB = x ∈ IR|ZU(x,C)
β(0, B(y))−2 dy = ∞, ∀C > 0;
MB = x ∈ IR|ZU(x,C)
β(B(y), 0)−2 dy = ∞, ∀C > 0;
NB = x ∈ IR|0 ∈ B(x); NB = x ∈ IR|B(x) = 0.
Theorem 1 Stochastic differential inclusion (1) has selectionwise unique in lawexplicit weak solutions with respect to every Borel measurable selection for everyinitial distribution if and only if
MB ⊆NB and NB ⊆MB .
Theorem 2 IfMB ⊆NB and NB ⊆MB ,
then there exists explicit selection v such that stochastic differential inclusion (1)has unique in law with respect to the selection explicit weak solutions for everyinitial distribution.
Theorem 3 For every initial distribution, stochastic differential inclusion (1) hasunique in law non-trivial explicit weak solution if and only if
NB =MB = ∅
and mapping B is almost everywhere single-valued.
References
[1] Kisielewicz M., Michta M., Motyl J, Set valued approach to stochastic control.I. Existence and regularity properties Dynam. Systems Appl. —2003. —Vol. 12,No 3-4. —P.405—- 431.
[2] Levakov A.A., Stochastic differential inclusions. Differential equations. —1997.—Vol.33,No 2. —P.212—220.
[3] Lepeyev A.N., Stochastic differential inclusions with unbounded diffusion co-efficient. Proceedings of the National Academy of Sciences of Belarus, Physicaland mathematical sciences series. —2005. —No 3. —P.37—43.
Belarusian State University,pr. Nezavisimosti 4a, 220030 Minsk Belaruse-mail: [email protected]
183
Methods to compute Bayes classifiersPavel Lukashevich and Boris Zalesky, Belarus
For the last years several approaches have been developed to compute Bayesestimators. Combinatorial methods that allow computation of high dimensionalmaximum a posteriori estimates are among them.
We present results that were specially proposed to solve practical problems ofcomputation of maximum a posteriori estimates. They are based on new integerprogramming and combinatorial optimization methods and enable exact compu-tation of the Gibbs and Markov Random Fields estimators, if they are functionsof many integer variables and those functions are representable by submodularBoolean polynomials. For that the special techniques have been described to re-duce problems of integer minimization to the problem of minimization of Booleanpolynomials.
The Modified Submodular Function Minimization (MSFM) algorithm was de-rived to adapt for this purpose the Submodular Function Minimization algorithm[1]. The MSFM-algorithm is especially efficient to minimize submodular Booleanpolynomials arising from the Gibbs and Markov Random Fields, since usuallythese polynomials have local dependence of variables. The algorithm allows notonly to reduce significantly number of operations to compute the Gibbs andMarkov Random Fields estimators but also to perform computations in a con-currency mode.
We also present a new Bayes classifier that uses probabilistic and geomet-rical description of multidimensional data. It is supposed that we are given alearning sample of random variables X1, X2, . . . , Xn ∈ Rd accompanied with ateacher classification Y1, Y2, . . . , Yn , Yi ∈ 0, 1 . The random variables Xi aresupposed to be results of observation with random errors of two nonintersect-ing sets Ω0,Ω1 . The sets Ω0,Ω1 are supposed to be connected and bounded bysmooth curves. In addition, it is assumed that the Bayes density of each Xi isone of f`(x) =
RΩ`f(x,y)dy, ` = 0, 1. On standard conditions for f(x,y) and
forms of boundaries of the sets it is proven that the ML estimate, which maxi-mizes the joint density of all Xi over sets Ω0,Ω1 with smooth boundaries, areconsistent so that estimators bΩ`,n → Ω` , n→∞ .
The research was supported by INTAS grant No. 04-77-7036.
References
[1] S. Iwata, L. Fleischer, S. Fujishige. A Combinatorial, Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions. RIMS Kokyuroku,1120(1999), p.11-23.
United Institute of Informatics Problems,National Academy of Sciences of Belarus,Surganova 6, 220012 Minsk, Belaruse-mail: [email protected]
184
Computation the intensity - duration -frequency curves for the western Sub
Carpathians catchmentCarmen Maftei and Alina Barbulescu, Romania
Dimensioning of projects concerning hydraulic structures or water workprojects implies the cognizance of the design flood, i.e. the flood hydrograph as-sociated with a return period (frequency). A critical rainfall event that is usedfor assessing the flood hydrograph of a certain return period is called "designrainfall".
There are two ways of approaching this problem to find acceptable solutions.- To simulate time series of rainfall in compliance with the statistical behavior
of the ensemble of the defining elements. This approach is more correct fromconceptual point of view but it demands reliable data on long time series of rainfall.
- To neglect the influence of the variability of one or more characteristicsthat define the rainstorm upon the rainfall-runoff model, or to consider as realstrong relationships between some such defining elements (for example the meanintensity-duration). Both approaches belonging to the first way are simpler andlead to the elaboration of a set of procedures named the "event simulation" thatwill be considered further on.
For small basins the most used method for estimating the quartile of themaximum annual discharge starting from the rainfall intensity is the "RationalMethod". According to this formula the rainfall intensity is considered for a du-ration that is at least equal to the time of concentration of the basin. This meansthat for punctual rainstorms a relationship between the intensity - duration -frequency (IDF curves) has to be established.
Design storms that are derived from IDF curves are based on a main assump-tion that the internal intensities of rainfall throughout its duration have the samefrequency.
The purpose of this study is mainly to produce IDF-curves for precipitation fora small Romanian watershed. The Voinesti catchment is a representative water-shed for the western Sub Carpathians Mountain. The rainfall series data measuredthis catchment consist in a 126 events. More physically based models for the IDF-curves are proposed. The IDF-curves for this region are an interesting tool to beused in sewer system design to combat the frequently occurring inundations insemi-urbanized and urbanized areas of the Voinesti region.
References1. B. Mohymont, G. R. Demaree, D. N. Faka, Establishment of IDF-curves for pre-
cipitation in the tropical area of Central Africa - comparison of techniques andresults, Natural Hazards and Earth System Sciences (2004) 4: 375-387
2. C. Maftei, Modelisation spatialisee de l’ecoulement sur des petits basins versantsEd. Cermi, Iasi, 2004
3. P. Meyan, A. Musy, Hydrologie frequentielle, Editions HGA, Bucarest, 1999
”Ovidius” University of Constantza, Bd.Mamaia 124, 900527,Constantza, Romania, e-mail: [email protected]
185
Mixture of regressions: nonparametricapproach
Rostislav Maiboroda, Ukraine
Nonparameric models are quite unusual in the statistics of finite mixtures. It iswell known that such models are, generally speaking, inidentfiable. Hall and Zhou(2003) considered a two-component multivariate model at which the variables areindependent for each component. This models seems to be a unique example ofidentifiable nonparametric finite mixture problem for i.i.d. observations. In thistalk we discuss a model which allows to analyze dependence of variables in two-component bivariate mixture.
We consider bivariate i.i.d. observations (Xi, Yi)ni=1 from a mixture of twocomponents:
PXi < x, Yi < y = pH1(x, y) + (1− p)H2(x, y),
where the distributions of components Hi are described by the regression model
Y = gi(X) + ε,
gi are unknown functions, ε is a random error independent of X with CDF Fsymmetric around zero. (F is the same for both components)
To start with, assume that gi(x) = bi doesn’t depend on x . Then the modelturns to
PYi < y = pF (y − b1) + (1− p)F (y − b2).
The estimation of b1 , b2 and p is an identifiable problem in this model if b1 6 b2 and0 < p < 1/2 . It can be solved by equating first six empirical moments of Yini=1 tothe theoretical moments derived from the model. The moment estimator obtainedby this procedure is consistent under natural assumptions.
If gi(x) are smooth enough functions of x then they can be estimated bythe moment estimator in which the empirical moments evaluated by all thesample Yini=1 are replaced by some nonparametric regression estimates ofE(Y k | X = x) , e.g. of Nadaraya-Watson type.
Asymptotic properties of such estimators will be discussed.
References
1.Hall P., Zhou X.-H. (2003) Nonparametric estimation of component distrib-utions in a multivariate mixture. Ann.Statist., V. 31, No 1, 201-224.
Kyiv National Universitye-mail: mre@ univ.kiev.ua
186
Preservation of the Convergence of Solutionsof Stochastic Equations
S. Makhno, Ukraine
We consider solutions of Ito stochastic equations whose coefficients depend ona small parameter ε > 0 .
ξε(t) = x+
Z t
0
bε(s, ξε(s))ds+
Z t
0
σε(s, ξε(s)dw(s).
We investigate their convergence as ε → 0 without assuming the convergenceof the coefficients themselves. Conditions under which the solutions convergenceas ε → 0 in the weak sense to solution or Ito stochastic equation or stochasticequation with a local time was obtained in the author works. Now we perturb theoriginal coefficients by functions that also depend on a small parameter ε and con-sider solutions of the stochastic equations with perturbed coefficients. What can besaid about the convergence of their solutions? In the case of the sufficient smooth-ness of the coefficients and the uniform convergence of the perturbing functions tozero, an answer to this equation follows from the theorem on integral continuityof solutions with respect to the parameter: the limit process satisfies the sameequation as the process with unperturbed coefficients. In the report we consider asimilar problem in the case where the coefficients are not assumed to be smoothand depend irregularly on a small parameter. Moreover, the perturbing functionsmay not be "small" at certain points and tends to infinity as ε→ 0 or may havenot limit at all.
Institute for applied mathematics and mechanics,R.Luxemburgh str, 74, Donetsk, 83114, Ukrainee-mail: [email protected],ua
187
Efficiency comparison of two consistentestimators in a nonlinear errors-in-variables
regression model
Andrew Malenko, Ukraine
A nonlinear structural errors-in-variables model is considered, where the re-sponse variable Y has a conditional density of the form
f(y|ξ) = exp
yξ − C(ξ)
ϕ+ c(y, ϕ)
ff.
Here ξ = ξ(X,β) is the canonical parameter, X is latent scalar regressor, nor-mally distributed with parameters µ and σ2 (nuisance parameters which maybe known), ϕ > 0 means dispersion parameter, is known, β is unknown vectorto be estimated. Independent couples (Wi, Yi), i = 1, . . . , n are observed, whereW = X + U with U ∼ N (0, σ2
u) , independent of X .Two consistent asymptotically normal estimators of β are considered: Struc-
tural Quasi Score (SQS) and Corrected Score (CS). The comparison of efficiencyis studied in terms of asymptotic covariance matrices ACM of each estimator,ΣSQS and ΣCS . Two cases are considered separately: when nuisance parametersare known and when they are to be estimated.
The first case was studied mainly for small error variance, i.e. when σ2u → 0 .
It was shown [1] that ΣCS − ΣSQS = O(σ4u) . But in [2] the advantage of SQS
is shown for arbitrary σ2u . We consider the first case only to show expansions of
ΣSQS and ΣCS up to the order O(σ6u) and some consequences.
The second case of unknown nuisance parameters was partially studied in[3]. We use the expansions of the two ACM up to the order O(σ6
u) to show theadvantage of SQS estimator in the polynomial model for small σ2
u .
References
[1] A. Kukush, H. Schneeweiss and R. Wolf (2002). Comparing different estima-tors in a nonlinear measurement error model. Discussion paper No 244. SFB386, University of Munich.
[2] A. Kukush, H. Schneeweiss and S. Shklyar (2005). Quasi Score is more effi-cient than Corrected Score in a general nonlinear measurement error model.Discussion paper No 451. SFB 386, University of Munich.
[3] H. Schneeweiss (2006). The polynomial and the Poisson measurement errormodels: some further results on quasi score and corrected score estimation.Discussion paper No 446. SFB 386, University of Munich.
e-mail: [email protected]
188
Some aspects of stochastic calculus forfractional Brownian motion and close processes
Yuliya Mishura, Ukraine
We consider Wiener integrals with respect to fractional Brownian motion(fBm) and establish upper and lower bounds for their moments both on nonran-dom and random intervals. The Fubini theorem for Wiener and stochastic integralsw.r.t. fBm is proved. The existence of weak solution of stochastic differential equa-tion with nonsmooth drift and Wiener integral w.r.t. fBm is established. Strongsolutions for such equations are considered and their norms in various functionalspaces are estimated.
Statistical inference with fBm is developed. We study how to test whether afractional Brownian motion has a linear trend against a certain nonlinear trend.The origin of this problem is in the financial modeling with fBm. Some relatedquestions, like goodness-of-fit test and volatility estimation in these models, arestudied.
We consider mixed Brownian–fractional Brownian model (mBfBm) and estab-lish the absence of arbitrage in the Black–Scholes setup with Markov strategies.
Filtering problems are considered in the most extensive framework, when bothobservable and hidden processes are presented by mBfBm. Solvable cases arestudied.
We construct the stochastic analysis of multiparameter fractional Brownianfields.
References
[1] Mishura, Yu., Nualart, D. Weak solutions for stochastic differential equationswith additive fractional noise. Stat. Probab. Lett. 70, No. 4, (2005), 253–261.
[2] Kukush, A.; Mishura, Yu., Valkeila, E. Statistical inference with fractionalBrownian motion. Stat. Inference Stoch. Process. 8, No. 1, (2005), 71–93.
[3] Mishura, Yu. Fractional stochastic integration and Black-Scholes equation forfractional Brownian model with stochastic volatility. Stoch. Stoch. Rep. 76,No. 4, (2004), 363–381.
[4] Il’chenko, S. A., Mishura, Yu.S. Generalized two-parameter Lebesgue-Stieltjesintegrals and their applications to fractional Brownian fields. Ukr. Mat. Zh. 56,No. 4, (2004), 435–450.
[5] Mishura, Yu.S. Quasi-linear stochastic differential equations with fractionalBrownian component. Teor. Jmovirn. Mat. Stat. 68, (2003), 95–106.
Department of Mechanics and Mathematics,Kyiv National Taras Shevchenko University,64 Volodymyrska, 01033 Kyiv, Ukrainee-mail: [email protected]
189
The problem of arbitrage in the mixed market.Approximation of the fractional Brownian
motion.Yuliya Mishura, Taras Androshchuk, Ukraine
The mixed Brownian – fractional Brownian version of the Black-Merton-Scholes model is considered, i.e., a
`B,S
´-market with a bond B and a stock
S , where
Bt = ert, St = eaWt+bBHt +ct, r, a, b, c ∈ R, a, b 6= 0. (1)
We restrict ourselves with strategies π = (βt, γt)t≥0 which are1) self-financing:`
Bt · βt + St · γt´−`B0 · β0 + S0 · γ0
´=
Z t
0
βs dBs +
Z t
0
γs dSs;
2) of Markov type:
βt = β`St, t
´, γt = γ
`St, t
´. (2)
The main result we establish is that there is no arbitrage opportunity in theclass of self-financing Markov-type strategies with sufficiently smooth β(·, ·), γ(·, ·)in (2). To prove this we give a characterization of the self-financing property forthe Markov-type strategies. It turns out that under condition that the supportof the distribution of St coincides with [0,+∞) for all t > 0 , the process ofcapital Xt = Bt ·βt +St · γt constructed by a self-financing Markov-type strategynecessarily has a form Xt = φ(St, t) , where function φ satisfies parabolic equation
φ′t(x, t) +a2
2x2 φ′′xx(x, t) + r xφ′x(x, t)− r φ(x, t) = 0.
The analyze of this equation gives that φ ≡ 0 if only the necessary conditions ofarbitrage X0 = 0 and XT ≥ 0 (a.s.) hold, which in turn means that the arbitrageis impossible.
The next result we give is the representing of the process of capital in themixed model (1) as a limit of certain semimartingales. For this aim we in-troduce the process BH,α which is continuously differentiable for α > 0 andfor which the convergence E
˛BHt −BH,αt
˛m→ 0 holds for 2 ≤ m < 1
1−Has α → 0+ . Also we prove that if the integrand f almost surely belongs tothe class C2(1−H)+ε with some ε > 0 , then for H ∈
`34, 1´
convergence in prob-ability
R T0f(u) dBH,αu →
R T0f(u) dBHu holds as α→ 0+ , where the last integral
is the Riemann-Stieltjes one. The sequence of semimartingales which converges tothe process of capital is built using BH,α .
Kyiv National Taras Shevchenko University,Department of Mechanics and Mathematics,Volodimirska, 64, 01033, Kyiv, Ukrainee-mail: [email protected], [email protected]
190
Fractal approximation of functions in somemetric spaces and its application to image
coding problemDmitry Yu. Mitin, Nikolai A. Nazarenko, Ukraine
Classical papers that began fractal methods of lossy image compression pro-posed some sufficient conditions for fractal transform operator, defined either inspace of plane subsets or in space of functions, to be a contraction [1]. It waslater shown that the contractivity conditions for fractal operator were too re-strictive since algorithms of fractal approximation sometimes converge even whenthese conditions are not fulfilled. In many cases the problem of finding sufficient(or necessary and sufficient) conditions for convergence of the fractal transformiterations, neither too restrictive nor too complicated to be checked, remains in-vestigated in part only.
One of approaches usually proposed [2, 3] is to weaken the contractivity condi-tion to so called eventual contractivity one, i. e., the contractivity for large enoughpowers of the fractal operator.
Recall some definitions [4]. Given function f bounded in segment I ⊂ R ,by G(f) denote an extended graph of f , i. e., the minimal closed con-vex, with respect to the second coordinate, set containing graph of f ;G(f) = (x, y) : y ∈ [I(f, x),S(f, x)] , x ∈ I , G(f, J) = (x, y) ∈ G(f) : x ∈ J ,J ⊂ I . Here I(f, x) = limδ→+0 I(f, x, δ) , I(f, x, δ) = inf |y−x|<δ f(y) ,S(f, x) = limδ→+0 S(f, x, δ) , S(f, x, δ) = sup|y−x|<δ f(y) . Let us identify func-tions with the same extended graphs. Hausdorff distance between functions f angg is defined as:
hα,J(f, g) = max( supu∈G(f,J)
infv∈G(g,J)
ρα(u,v), supv∈G(g,J)
infu∈G(f,J)
ρα(u,v)), α > 0,
ρα((u1, u2), (v1, v2)) = max(α−1|u1 − v1|, |u2 − v2|) ,hα,I(f, g) = hα(f, g) . By definiion, Hausdorff modulus of continuity isωHα (f, δ) = hα(I(f, · , δ/2),S(f, · , δ/2)) . Function f is called continuous byHausdorff (f ∈ HC(I)) if ωHα (f, δ) → 0 as δ → +0 .
By HCα,Ω(I) denote the set of functions f ∈ HC(I) with property:ωHα (f, δ) ≤ Ω(δ) , where Ω(δ) ↓ 0 as δ → +0 . It is known [4] that (HCα,Ω(I), hα)is a complete metric space.
Let us given segments I, I1, . . . , In, I′1, . . . , I
′n such that
Sni=1 Ii = I ,
Ii ∩ Ij = ∅ ( i 6= j ), I ′i ⊂ I and homeomorphisms (ϕi(x), ψi(x, y)) , x ∈ I ′i ,y ∈ R , ϕi(I
′i) = Ii , i = 1, . . . , n , with |ϕi(x1) − ϕi(x2)| ≤ d′i|x1 − x2| ,
|ψi(x1, y1) − ψi(x2, y2)| ≤ d′′i ρα((x1, y1), (x2, y2)) , x1, x2 ∈ I ′i , y1, y2 ∈ R ,max(d′i, d
′′i ) =: di .
Put fractal transform operator [1] equal to:
(T (f))(x) =
nXi=1
ψi(ϕ−1i (x), f(ϕ−1
i (x)))1IIi(x), x ∈ I.
191
Consider a closed set F ⊂ HCα,Ω(I) such that T (F) ⊂ F .Let
limk→∞
“max
1≤i1,...,ik≤ndi1 . . . dikγi1,...,ik
”1/k
< 1, (∗)
where
γi1,...,ik = supf,g∈Ff 6=g
ˆhα, ϕ−1
i1(Ii1∩ϕ
−1i2
(Ii2∩...ϕ−1ik
(Iik)))
(f, g) · (hα(f, g))−1˜.Theorem 1 Under suppositions made, there exists a unique, up to the equiva-lence, function f∗T ∈ F , which is the fixed point of the operator T . For any f ∈ F
one has T k(f) → f∗T as k →∞ .
Thus inequality (∗) is the eventual contractivity condition for the operatorT .
The question of convergence for the fractal transform iterations is studied indifferent metric spaces, e. g. spaces with such distances: uniform [2, 3], average ofsome order [3], Hausdorff [4, 5], integral Hausdorff [4], Kantorovich-Wasserstein-Monge etc.
In practice, function f (in two-dimensional case) can have the sense of greycolor intensity for greyscale image. The main idea of fractal image compression isto code the function f∗T (or one that is close enough to it) by the vector of theoperator T parameters.
1. Barnsley M. F., Hurd L. P. Fractal image compression. — Wellesley: A. K.Peters, 1993.
2. Vasil’yev S.N. Methods of fractal interpolation of Barnsley type // RussianMathematics 46 (2002), N 9, p. 1–12.
3. Mitin D. Yu., Nazarenko M. O. Fractal approximation in spaces C and Lpand its applications to image coding problems // Problems of the theory ofapproximation of functions and related problems: Collection of works of theInstitute of Mathematics of Ukrainian Academy of Sciences. — Kyiv, 2006.(In Ukrainian, to be published.)
4. Sendov Bl. Mathematical modeling of real-world images // ConstructiveApproximation 12 (1996), N 1, p. 31–65.
5. Mitin D. Yu., Nazarenko N.A. Fractal approximation of functions in somemetric spaces // Modern problems of the theory of functions and their appli-cations: Abstracts of the 13th Saratov Winter School. — Saratov: NauchnayaKniga, 2006. — P. 116–117. (In Russian.)
Taras Shevchenko Kyiv National University,Faculty of Mechanics and Mathematics,Department of Mathematical Analysise-mail: [email protected], [email protected]
192
Filtering problem for multidimensionalstationary sequences
Mikhail Moklyachuk and Aleksandr Masyutka, Ukraine
We deal with the problem of optimal linear estimation of the functionalAξ =
P∞j=0 a(j)ξ(−j) which depends on the unknown values of a multidimen-
sional stationary sequence ξ(j) = ξk(j)Tk=1 with spectral density matrix F (λ)
from observations of the sequence ξ(j)+η(j) for j ≤ 0 , where η(j) = ηk(j)Tk=1
is a multidimensional stationary sequence uncorrelated with ξ(j) with spectraldensity matrix G(λ) .
The sequence ξ(j)+η(j) admits the canonical moving average representation
ξ(j) + η(j) =
jXu=−∞
d(j − u)ε(u) (1)
if the spectral density F (λ) + G(λ) = fij(λ) + gij(λ)Ti,j=1 of the stationarysequence ξ(j) + η(j) admits the canonical factorization:
F (λ) +G(λ) = d(λ)d∗(λ), d(λ) =
∞Xk=0
d(k)e−ikλ, (2)
where d(k) = dij(k)j=1,m
i=1,T, ε(u) = εk(u)mk=1 is a multidimensional white noise
sequence: E |εk(u)|2 = 1, k = 1,m, Eεi(t)εj(s) = 0, t 6= s . Regularity of spectraldensity matrices F (λ) , G(λ) is the sufficient condition for the factorization (2).Regular spectral densities F (λ) , G(λ) admit the canonical factorizations:
F (λ) = ϕ(λ)ϕ∗(λ), ϕ(λ) =
∞Xk=0
ϕ(k)e−ikλ, ϕ(k) = ϕij(k)j=1,m
i=1,T; (3)
G(λ) = ψ(λ)ψ∗(λ), ψ(λ) =
∞Xk=0
ψ(k)e−ikλ, ψ(k) = ψij(k)j=1,m
i=1,T. (4)
The value of the mean square error ∆(h;F,G) of a linear estimate∧A ξ of the
functional Aξ is determined by the formula
∆(h;F,G) = E
˛Aξ −
∧A ξ
˛2= ‖Ψa‖2+‖D(a− h)‖2−〈Ψ(a− h),Ψa〉−〈Ψa,Ψ(a− h)〉 ,
‖Ψa‖2 =∞Pk=0
‖(Ψa)k‖2, (Ψa)k =kPl=0
a(l)ψ(k − l); ‖D(a− h)‖2 =∞Pk=0
‖(D(a− h))k‖2,
(D(a−h))k =kPl=0
(a(l)− h(l))d(k − l); 〈Ψ(a− h),Ψa〉 =∞Pk=0
〈(Ψ(a− h))k, (Ψa)k〉
where h(λ) =∞Pk=0
h(k)e−ikλ is the spectral characteristics of the estimate∧A ξ .
193
The spectral characteristics h(F,G) of the optimal linear estimate is deter-mined by the condition:
∆(F,G) = ∆(h(F,G);F,G) = minh∈L−2 (F+G)
∆(h;F,G), (5)
where L−2 (F +G) is a subspace of the space L2(F +G) generated by functions:einλδk , δk = δklTl=1 , k = 1, T , n < 0 , where δkk = 1, δkl = 0 for k 6= l .
In the case where the densities admit factorizations (2), (4), the mean square
error of the optimal estimate∧A ξ can be calculated by the formula
∆(F,G) = ‖Ψa‖2 − ‖B∗Ψ∗Ψa‖2 = 〈cG, a〉 − ‖CGb∗‖2 , (6)where
cG(k) = (Ψ∗Ψa)k =∞Pl=0
(Ψa)l+kψ∗(l), (B∗Ψ∗Ψa)k =
∞Pl=0
(Ψ∗Ψa)l+kb(l)∗,
(CGb∗)k =
∞Pl=0
cG(l + k)b∗(l).
Here b(λ) = bij(λ)j=1,T
i=1,mis a matrix function which satisfies the equation
b(λ)d(λ) = Im, where Im is the identity matrix of order m.The spectral characteristics h(F,G) of the optimal estimate is equal to
h(F,G) = A(eiλ)− rG(eiλ)b (λ), rG(eiλ) =
∞Xk=0
(CGb∗)ke
−ikλ. (7)
If the densities admit the canonical factorizations (2), (3), then the meansquare error and the spectral characteristics of the optimal estimate are deter-mined by the formulas:
∆(F,G) = 〈cF , a〉 − ‖CF b∗‖2 , (8)
h(F,G) = rF (eiλ)b(λ), rF (eiλ) =
∞Xk=0
(CF b∗)ke
−ikλ, (9)
where cF (k) = (Φ∗Φa)k =∞Pl=0
(Φa)l+kϕ∗(l) , (Φa)k =
kPl=0
a(l)ϕ(k − l) .
Theorem 1 The value of the mean square error ∆(F,G) of the optimal linearestimate of the functional Aξ which depends on the unknown values of a multidi-mensional stationary sequence ξ(j) from observations of the sequence ξ(j) + η(j)for j ≤ 0 , where ξ(j) , η(j) are uncorrelated multidimensional stationary se-quences which have densities F (λ) , G(λ) that admit canonical factorizations (2),(4) or (2), (3), can be calculated by the formulas (6), (8). The spectral character-istics h(F,G) of the optimal estimate is determined by the formulas (7), (9).
Kyiv Taras Shevchenko University,Mechanics and Mathematics Faculty,Kyiv 01033, Ukrainee-mail: [email protected]
194
Behaviour of LSE in long-memory regressionmodels with restrictions
Elina Moldavskaya, Ukraine
We examine the solution of minimization problem of the least square functionalof linear regression with long-memory and inequality constraints on parameters.It is proved that this solution, normed with an appropriate matrix, tends to thesolution of the quadratic programming problem. However this solution is a non-Gaussian random vector in the typical cases. It is as against known results withoutconstraints, where the same transformation has a normal asymptotic distributionin the typical cases.
References
[1] Ivanov A.V., Leonenko N.N. Asymptotic behavior of M-estimators incontinuous-time non-linar regression with long-range dependent errors//Random oper. and Stoch. Equ.-2002.-Vol.10,N 3.-P. 201-222.
[2] Korkhin A.S. About some properties of regression parameters estimators un-der apriori inequality-constraints// Kibernetika.-1985.-Vol. 6.-P. 106-115.
[3] Leonenko N.N and Moldavskaya E.M. Non-Gaussian scenarios in long-memory regression models with non-linear constraints // Reports of the NASof Ukraine .-2002.-N 2.-P.44-46.
[4] Moldavskaya E.M.Asymptotic distributions of non-linear inequality con-strained least squares estimation of linear regression coefficients in modelswith strong dependence//Theory Prob. and Math. Stat.-2006 (to appear)
[5] Taqqu M.S. Convergence to intagrated process of arbitrary Hermite rank //Z. Wahrshein. verw. Gebiete. –1979. –Bd 50 –P.53–83.
Institute of Cybernetics NAS of Ukraine,Glushkov av., 40, Kiev 03680, Ukraine
Department of Mathematics,Ben-Gurion University of the Negev,P.O.B. 653, Beer-Sheva 84105, Israel
e-mail: [email protected]
195
Strong stability in the (R,S) inventoryproduction model
Mouhoubi Zahir, Algeria, Aissani Djamil, Algeria
The strong stability investigation allow us to obtain the error of approximationof the stationary and non stationary characteristics for two inventory-productionbackorder processes J and eJ considered in (R,S) inventory-production models(Σ) (ideal model) and (eΣ) (real model) under different order demand probabil-ity distribution and under assumption that the norm deviation of the two orderdemand distribution is small with respect to a given norm (closeness of the proba-bility distribution of the second demand process to the probability distribution ofthe the first one). In other word, we are interesting about the effect of perturbationof the order total demand distribution on the behavior of the inventory-productionbackorder process.
Tne inventory backorder process J is a an homogeneous Markovian processwith a phase space (E, E) where E = . . . ,−2,−1, 0, 1, . . . , S and E is the σ -algebra generated by the singletons of E with S is a fixed integer which representsthe finite capacity of stock. We have showed that the Markov process is ergodicunder the following condition
c > E(D). (1)Let us introduce the v -norm in the finite measure mE space defined as follows‖µ‖β =
Pj∈E v(j)|µj | where v(j) = βj with β is a parameter such that β > 1 .
Notice that all notations and definitions used in this abstact are specified in refer-ences [1,3,4,5 ]. Thus we have clarified the strong stability condition given by thefollowing result. For this, we denote dn, n, 1, . . . the probability distribution of thedemand process D , π the stationary probability distribution of the Markovianinventory backorder process J and c the fixed production capacity.
Theorem 1 The discrete inventory backorder process J in (R,S) inven-tory/production model with limited capacity, uncertainty demand and random lead-time is uniformly ergodic, strongly stable and aperiodic, under the condition (1),with respect to the norm ‖.‖β for all β ∈]1, ω[ where ω verifie E(ωD−c) = 1 .
Theorem 2 We consider for all real number Υ > 1 such that ρ(β)Υ < 1 andΥ < κ−1 . Then, we have
‖P t −Π‖β ≤ ρ(β, c)t + βcρ(β)Υ−t
(1−ρΥ)2maxd−1
0 πS ,Λ(Υ)
with Λ(Υ) = sups≥0 |λ(s) − d−10 πS |Υs , πS is the S -th component of the unique
stationary probability distribution π of the chain J and ρ(β, c) = E(βD)βc .
Let us introduce the following proximity measure of the probability distributiondk, n = 0, 1, . . . and edk, n = 0, 1, . . . of amount demand in each period for the twoinventory production models (Σ) and (eΣ) as follows
W((dk), (edk)) =
+∞Xl=0
βl|dl − edl| (2)
196
Theorem 3 Assume that P and eP is the transition kernels of the Markov chainsJ and eJ respectively. Then, the deviation between the transition kernels P andeP is given as below
‖P − eP‖β = W((dk), (edk)). (3)
For convenience, we use the following notation W((dk)k, (edk)k) = W in the sequel.Let us denote by γ = max(E(D),E( eD)) and [x] the smallest integer not less
than x− 1 . Under conditions of theorem and assume that it exits β ∈]1, ω[ suchthat W((dk), (edk)) . Then, for all θ > γ it exists c0 such that for all
c ≥ c0 = [θ − ln(1−W((dk), (edk)))
lnβ] + 1. (4)
Then, we have the following inequality holds
supt≥0
‖ eP t − P t‖β ≤ HW((dk), (edk)) (5)
where H = 1
(1−ρ(β,c)−W((dk),( edk)))2
„1
1−ρ(β,c) + βcρ(β,c)Λ0(1−ρ(β,c))2 +
2d−10 πSβ
cρ2(β,c)
(1−ρ(β,c))3
«.
In the stationary case, we establish the deviation of the stationary probabilitydistributions π and eπ of the Markov chains J and eJ respectively.
Theorem 4
‖π − eπ‖ ≤ βc−Sρ(β, c)
(1− ρ(β, c)−HW((dk), (edk)))2(1 +
βc−Sρ(β, c)
1− ρ(β, c))HW((dk), (edk)) (6)
Remark 1 Notice that when HW((dk), (edk)) −→ 0 then supt≥0 ‖ eP t−P t‖β −→ 0and ‖π − eπ‖β −→ 0 .
More results are obtained and analyzed which will be the subject of the fullpaper.
References[1] Aissani, D., Kartashov, N.V., 1983. Ergodicity and stability of Markov chains with
respect to operator topology in the space of transition kernels. Doklady AkademiiNauk Ukrainskoi SSR seriya A 11, 3-5.
[2] Borovkov, A.A., 1998. Ergodicity and Stability of Stochastic Processes. WileySeries in Probability. John Wiley and Sons Ltd.
[3] Kartashov, N.V., 1996. Strong Stable Markov Chains. TBIMC Scientific Publish-ers, VSP, Utrecht.
[4] Mouhoubi, Z., Aissani, D., 2005. Some Inequalities of the Uniform Ergodicity andStrong Stability of Homogeneous Markov Chains. Pliska Stud. Math. Bulgar. 17,171-186.
[5] Mouhoubi, Z., Aissani, D., 2003. On the Uniform Ergodicity and Strong StabilityEstimates of Waiting Process. Bulletin of the International Statistical Institute,Volume LX, Book 2, 97-98.
Laboratory of Modelization and Optimization of SystemsFaculty of Sciences and Engineer SciencesUniversity A.Mira of Bejaia (06000), Algeriae-mail: [email protected]
197
The Laplace transformation of the ergodicdistribution of the process semi-markov
random walk with negative drift, pozitivejumps, latings and with delaying screen in the
zeroT.I.Nasirova, E.A.Ibayev, T.A.Aliyeva, Azerbaijan
Let on the probability space (Ω, F, P (.)) is given the sequence,ξk(ω), ηk(ω), ζk(ω)∞k=1 , where (ξk(ω) ,ηk(ω), ζk(ω)) ,k = 1,∞ , are identicallydistributed, independents and independents themselves, ξk(ω) > 0, ηk(ω) > 0 ,ζk(ω) > 0, k = 1,∞ .
We construct the process
X1(t, ω) =
8>>>>>>>>>><>>>>>>>>>>:
z − t+k−1Pi=1
[ζi (ω) + ηi(ω)] , ifk−1Pi=1
[ξi(ω) + ηi(ω)] ≤ t <
<k−1Pi=1
[ξi(ω) + ηi(ω)] + ξk(ω),
z −k−1Pi=1
[ξi(ω) + ηi(ω)] , ifk−1Pi=1
[ξi(ω) + ηi(ω)] + ξk(ω) ≤ t <
<kPi=1
[ξi(ω) + ηi(ω)] , z > 0.
This process we shall call the process of semi-markov random walk with negativedrift, pozitive jumps and with latings.
We delay process X1 (t, ω) with screen in the zero:
X (t, ω) = X1 (t, ω)− inf0≤s≤t
(0, X1 (s, ω)) .
Our aim to find the Laplace transformation of the ergodic distribution of theprocess X (t, ω) .
We denoteR (t, x) = Pω : X (t, ω) < x, x > 0,
R (t, x|z ) = Pω : X(t, ω) < x |X (0, ω) = z ,
R( θ, x ) =
∞Zt=0
e−θ tR(t, x ) dt, θ > 0, R( θ, x | z ) =
∞Zt=0
e−θ tR(t, x | z ) dt ,
˜R( θ, α ) =
∞Zx=0
e−αxdxR(θ, x ), α > 0, ˜R( θ, α | z ) =
∞Zx=0
e−αxdxR(θ, x | z ),
˜R( θ, α) =
∞Zz=0
˜R(θ, α | z ) dPX(0, ω) < z , R(α) = limθ→0
θ ˜R (θ, α) = E e−αX(ω),
where X (ω) - the limit random variable obtained from process X (t, ω) in thesince ergodic.
With the purpose of a findingR(α)we must find ˜R( θ, α | z ).
198
Theorem 1 The ˜R( θ, α | z ) is satisfies the following integral equation
˜R(θ, α | z) = e−θ zzZ
x=0
e−(α−θ)xP ξ1(ω) > z − x dx+
∞Zt=z
e−θ tP ξ1(ω) > t dt+
+1− ϕ (θ)
θ
24−e−θ z zZα=0
e−(α−θ)xdxP ξ1(ω) < z − x+
∞Zt=z
e−θ tdP ξ1(ω) < t
35+
+ϕ (θ)
∞Zt=z
e−θ tdP ξ1(ω) < t ∞Z
y=0
˜R(θ, α | y) dP ζ1(ω) < y +
+ϕ(θ)
zZu=0
e−θ u
24 ∞Zy=z−u
˜R(θ, α | y)dyP ζ1(ω) < y − z + u ] dPξ1(ω) < u .
In the case
P ξ1(ω) < t =ˆ1− e−µ t
˜ε(t), µ > 0,
P ζ1(ω) < t =
"1− e−λ t
m+−1Pi=0
(λ t)i
i!
#ε(t), λ > 0,m+ = 1,∞
9>=>;,ε(t) =
0, t < 0,1, t > 0,
on the Tauber theorem we have
R(α) = limθ→0
θ ˜R (θ, α) =λ−m+µ
λ× (α+ λ)m
+×
×(m+(α− µ )λm+−1 − m+(m+ − 1)
2α(α− µ)λm
+−2
+λm+
+
m+Xi=3
Cim+(α− µ)αi−1λm+−i)−1.
From this form we find first two moments of the random variableX(ω) .
Baku State University,Institute of Cybernetics NASA,Azerbaijan State Economic Universitye-mail: [email protected]
199
The generation function of the distribution ofthe number for riching level "a" of the process
Nasirova T.H., Sadiqova R.H., Azerbaijan
Let on the probablity space (Ω,=, P (.)) is given the sequence ξk, ηkk=1,∞ ,where ξk, ηk , k = 1,∞ are i.i.d.r.v., ξk > 0 . We construct the process
X1(t) =m−1Pk=0
ηk, ifm−1Pk=0
ξk ≤ t <mPk=0
ξkm = 1,∞ , where η0 = z ≥ 0, ξ0 = 0.
We delay process X1(t) with screen in the zero. X2(t) = X1(t)− inf0≤s≤1
(0, X1(s)) .
Then we delay process with screen in the ”a” X(t) = X2(t)− sup0≤s≤1
(0, X2(s)− a).
We denote the number of the steps of the process X(t) the reaching the level awith νa1 (ω) . We denote through Ψ(u) = Euν
a1 (ω), 0 < u < 1 the generating
function of the distribution of the random variable νa1 (ω) .On total probability formula for expectation we find
(1) Ψ(u) = Euνa1 (ω) =
aRz=0
Ehuν
a1 (ω)|X(0, ω) = z
idP X(0, ω) < z
=aR
z=0
Ψ(u, z)dP X(0.ω) < z .
By total probability formula we have(2) Ψ (u, z) = uP z + η1 > a+u
R ay=0
Ψ(u, y)P z + η1 < a; max(0, z + η1) ∈ dy .
The equation (2) we shall solve in the class of the complex Laplas distribution.
Fη−+1 +η+
2 +η−1(t) = P η1 < t =
8<:λ2
(λ+µ)2, t < 0,
1− µλ+µ
h1 + λ
λ+µ+ λt
ie−λt t > 0.
P η1 > t =
8<: 1− λ2
(λ+µ)2eµt, t < 0,
µλ+µ
h1 + λ
λ+µ+ λt
ie−λt t > 0.
Then the integral equation will be written in following form:
(3) Ψ(u, z) = u µλ+µ
[1+ λλ+µ
+λ(a−z)]e−λ(a−z) +u λ2µ(λ+µ)2
e−µzzR
y=0
Ψ(u, y)eµydy+
+u λ2µ(λ+µ)2
eλzaR
y=0
Ψ(u, y)e−λydy−uλ2µzλ+µ
eλzaR
y=0
Ψ(u, y)e−λydy+
+λ2µuλ+µ
eλzaR
y=z
yΨ(u, y)e−λydy + λ2µ(λ+µ)2
Ψ(u, 0)e−µz.
From (3) we receive the differential equation .(4) Ψ′′′
z (u, z)− (2λ− µ)Ψ′′z (u, z) + λ(λ− 2µ)Ψ′
z(u, z) + λ2µ(1− u)Ψ(u, z) = 0.To solve the differential equation (4) we find from integral equation the bound-ary conditions. From received the solution helping form (1) we find Ψ(u) . Forapplications we find the expectation and variance of the distribution of νa1 (ω) .
Institute of Cybernetics, Azerbaijan National Academy of Sciences,F.Agayev st.9, Baku AZ1141, Baku; e-mail: [email protected]
200
A successive approximation method forsolution of actuarial integral equations
Bogdan Norkin, Ukraine
We consider various generalizations of the classical risk process (Kramer -Lundberg model) describing a stochastic evolution of the capital ξt of an insurancecompany, in particular, risk processes with deterministic premiums depending oncurrent capital [1], with random premiums [2], with non-Poisson flows of premiumsand claims [2], [3], risk processes in a stochastic markovian environment [4], [5].These processes satisfy the following stochastic integral equation
ξt = u+
Z t
0
c(ξs)ds+ Pt − Ct, t ≥ 0,
where u is an initial capital of the company, ξ0 = u ; c(·) is an intensity of deter-ministic (positive or negative) payments; Pt and Ct denote aggregated randompremiums and claims respectively.
For all processes we deduce integral equations for the probability of nonruinϕ(u) considered as a monotonic function of the initial capital u of the company.These equations have the following general form:
ϕ(u) = Aϕ(u), limu→+∞
ϕ(u) = 1,
where A is a corresponding linear integral operator depending on the process type(see [1]-[6] for details).
For a risk process in a Markovian environment a system of integral equationsfor probabilities of ruin from different initial states of the environment is obtained[4], [5].
General necessary and sufficient, and also concrete sufficient conditions forthe existence and uniqueness of solutions of the considered integral equations andsystems of equations are established. The necessary conditions require existenceof some monotonic functions ϕ∗(u) , ϕ∗(u) such that
0 ≤ ϕ∗(u) ≤ ϕ∗(u) ≤ 1, limu→+∞
ϕ∗(u) = 1,
Aϕ∗(u) ≥ ϕ∗(u), Aϕ∗(u) ≤ ϕ∗(u) ∀u ≥ 0.Here function ϕ∗(u) plays a role of generalized Cramer-Lundberg boundary. Suf-ficient conditions have a form of nonlinear inequalities for finding a generalizedLundberg constant. They are satisfied if a total average income of the companyis greater than its average expenditures in a unit of time.
We propose a successive approximation method for a numerical or analyticalsolution of the considered integral equations [1]–[6]:
ϕk+1(u) = Aϕk(u), ϕ∗(u) ≤ ϕ0(u) ≤ 1, k = 0, 1, . . . .
This method has been theoretically and practically validated, in particularits uniform convergence and rate of convergence has been established, numericalcalculations are presented. A technique for estimation of the accuracy of approxi-mations by approaching to the solution from below and from above is developed.The proposed method allows to calculate probability of ruin with any prescribedaccuracy.
201
[1] Norkin B.V. Method of successive approximations for solving integral equations ofthe theory of risk processes // Kibernetika i sistemnyi analiz. – 2004. – N 4. – P.10-18(In Russian, English translation in Cybernetics and systems analysis, Vol. 40 (2004),No. 4, P.517–526).
[2] Norkin B.V. Applications of the successive approximations method for finding theprobability of bankruptcy of an insurance company in case of stochastic premiums// Kibernetika i sistemnyi analiz. – 2006. – N 1. – P. 112-127 (In Russian, Englishtranslation in Cybernetics and systems analysis, Vol. 42 (2006), No.1).
[3] Norkin B.V. On Calculation of Probability of Bankruptcy for a Non-Poisson RiskProcess by the Method of Successive Approximations, // Problemy upravleniya iinformatiki. – 2005. – N 2. – P. 133-144 (In Russian, English translation in Journalof Automation and Information Sciences, Vol. 37 (2005), No. 4, P. 48–57).
[4] Norkin B.V. A system of integro-differential equations for the probability of bank-ruptcy of the risk process in a Markovian environment // Teoriya optimalnyh rishen.Issue 1. – Kiev: V.M.Glushkov Institute of Cybernetics of the National Academy ofSciences of Ukraine, 2002. – P. 21-29 (In Russian).
[5] Norkin B.V. The method of successive approximations for calculating the probabilityof bankruptcy of a risk process in a Markovian environment // Kibernetika i sistemnyianaliz. – 2004. – N 6. – P. 149-161 (In Russian, English translation in Cyberneticsand systems analysis, Vol. 40 (2004), No.6, P. 917-927).
[6] Norkin B.V. On a successive approximation method for calculation of the probabilityof bankruptcy of the classical risk process // Teoriya optimalnyh rishen. Issue 2. –Kiev: V.M.Glushkov Institute of Cybernetics of the National Academy of Sciences ofUkraine, 2003. – P. 10-18 (In Russian).
V.M.Glushkov Institute of Cybernetics of the NAS of UkraineProspect Academika Glushkova 40, 03680 Kiev-187, Ukrainee-mail: [email protected]
A new characterization of the normal law
Serguei Novak, UK
We present a solution to the inverse Shepp problem, and suggest a new char-acterization of the normal law that highlights a property of self-normalized ran-dom variables. We show also that a distribution is symmetric if and only if self-normalized random variables drawn from that distribution are uncorrelated.
Middlesex University Business School,London, United Kingdom
202
To a problem about diffusion processes whichsuppose generalized a vector of transposition
and a matrix of diffusionNovosjadlo A.F., Ukraine
Let’s consider two areas on a plane R2 : Di = x : x = (x1, x2) ∈ R2,(−1)ix2 > 0, i = 1, 2. Let’s designate boundary Di, i = 1, 2 as S . Let in D1 andD2 it is set the diffusion processes controlled by forming differential operators L1
and L2 accordingly:
Ll = 12
P2i,j=1 b
(l)ij
∂2
∂xi∂xj, l = 1, 2 ,
where numbers b(l)ij are elements of the symmetric positive definite matrixes Bl .Let’s set the task to describe wide enough class of the continuous Feller
processes on R2 , for which the forming operator coincides with Ll , l = 1, 2 inpoints of areas Dl . The given problem is termed as a problem of pasting togethertwo diffusion processes (see [1]). We apply analytical methods for its solution.At such approach required processes will be generated by a operator semigroupTt , t ≥ 0 , which is in turn defined through a solution of a corresponding problemof conjugation for linear uniformly parabolic second-order equation with explo-sive coefficients. Statement of a problem provides, that one of two requirements ofconjugation which are set on S , shows property requirements of process fellerityand the second requirement responds to common boundary condition of Ventsel[2] when in points of S such effects as diffusion and jump along the boundary arepossible, and also partial reflection. From the point of view of the theory of thepartial differential equations of parabolic type the considered problem of conju-gation has that singularity, that requirements of a joint coat are not satisfied (see[3]) and consequently, it can be referred only to conventionally correct. Classicalsolution of such problem of conjugation for the first time was gained by us by amethod of the limiting integral equations with use of a usual potential of a simplesphere. It is proved thus, that the Markovian process constructed by means of adiscovered semigroup can be treated as the generalized diffusion in understandingМ.I. Portenko (see [1], §1, ch. 3).
References
1. Portenko М.I. Processes of diffusion in media with membrans. - Kiev: Univer-sity mathematics NAS of Ukraine, 1995. -200p.2. Ventsel А.D. About boundary conditions for multidimensional diffusionprocesses. - The theory probability and it аpplication, 1959, v.4, 2, p. 172-185.3. Ivasishen S.D. Green’s Matrix of parabolic boundary problems. - Kiev: Thehight school, 1990.-200p.
Lviv, Universytetska, 1, LNU after Ivan Frankoe-mail: [email protected]
203
Actuarial Courses in Kyiv University andTraining Center for Actuaries and Financial
Analysts
Andriy Olenko, Ukraine
The courses "Contingencies"(Training center for actuaries and financial ana-lysts) and "Introduction to Actuarial Studies" (for Bachelor level, Kyiv university)were developed in the frame of Joint European TEMPUS PROJECTS.
The aim of these courses is to provide a grounding in the mathematical tech-niques which are of particular relevance to actuarial work in life insurance, pen-sions, health and care. The courses intend to give the students an insight into thelife insurance business and its institutional organization. These courses give stu-dents basic knowledge of life insurance mathematics, mortality theory and moregeneral stochastic processes in life insurance with applications to health insur-ance and multi-life insurance. Programs of the courses also include such topics:the equivalence principle, prospective reserves and differential equations for these,administration costs, gross premiums and premium reserves. Introduction to cred-ibility theory, stochastic models of insurance risk, the risk processes are taught inthe second part of the courses.
The courses are adapted to European standards in this area. Internet citesof the courses with lectures, problems, and another useful resources are devel-oped. One of the aims of these courses is students preparation to UK Faculty andInstitute of Actuaries exams.
Description, aims and objectives of the courses will be given in the presenta-tion. Overview of the courses programs, background readings and relevant actu-arial Internet resources will be presented. Various aspects of future developmentand improving of the courses will be discussed.
References
1. http://www.mechmat.univ.kiev.ua/probability/Actuarial Center/index.html
2. Olenko A. Actuarial mathematics. Problems. (second edition). Kyiv: KU-press. 67p. (2005).
Department of Probability Theory and Math. Statistics,Faculty of Mechanics and Mathematics,Kyiv University, Volodymyrska 64,01033, Kyiv, Ukraine
e-mail: [email protected]
204
Hyperbolic and Fractional HyperbolicBrownian motion with some applications
Enzo Orsingher, Italy
We study the diffusion on the Poincare half-plane model of a hyperbolic spacewith transition function pH = pH(x, y, t) satisfying the equation
∂pH∂t
=1
2y2
∂2
∂x2+
∂2
∂y2
ffpH (1)
with initial conditionpH(x, y, 0) = δ(x)δ(y − 1). (2)
This diffusion is called hyperbolic Brownian motion and describes the randommovement on a non-Euclidean space with negative curvature.
By means of the relationships transforming Cartesian coordinates into geodesicpolar coordinates(
x = sinh η cosαcosh η−sinh η sinα
y = 1cosh η−sinh η sinα
η > 0, 0 < α < 2π (3)
equation (1) is converted into
∂pH∂t
=1
2
∂2
∂η2+
1
tanh η
∂
∂η+
1
sinh2 η
∂2
∂α2
ffpH . (4)
After the time change t′ = t/2 and by deleting the dependence on α , we areable to prove that
pH(η, t′) =e−
t′4
√π“√
2t′”3
Z ∞
η
ϕe−ϕ2
4t′√
coshϕ− cosh ηdϕ, η > 0, t′ > 0. (5)
This formula (up to some constants) has been published without proof byGertsenshtein and Vasiliev in 1959.
A generalization of (4) and (5) is obtained by considering the fractional hy-perbolic equation
∂αpH∂tα
=1
2y2
∂2
∂x2+
∂2
∂y2
ffpH , 0 < α ≤ 1, (6)
subject to the initial condition
pH(x, y, 0) = δ(x)δ(y − 1). (7)
205
In this case the distribution at time t′ = t/2 of the process η = η(t′) , describedby the geodesic distance, reads
pH(η, t′) =2
π
Z ∞
0
xEα,1
„− t
′α
4− x2t′α
«dx
Z ∞
η
sinxϕ√2 coshϕ− 2 cosh η
dϕ, (8)
where Eα,1(x) is the Mittag-Leffler function.The stochastic representation of hyperbolic Brownian motion as(
X(t) =R t0eB2(s)− s
2 dB1(s)
Y (t) = eB2(t)− t2
(9)
and in terms of geodesic coordinates is examined.The analogy of hyperbolic Brownian motion with Brownian motion on the
sphere is discussed and analyzed by examining the corresponding stochastic dif-ferential equations. We show that hyperbolic Brownian motion can be thought ofas a Brownian motion on a sphere of imaginary radius and reversed time.
Some remarks concerning the exponential functionals (9) are presented andtheir financial interpretation is sketched.
References
Comtet A., Monthus C. (1996), Diffusion in a one-dimensional randommedium and hyperbolic Brownian motion, Journ. Physics A, Math. Gen. 29, 1331-1345.Gertsenshtein M.E., Vasiliev V.B. (1959), Waveguides with random inho-mogeneities and Brownian motion in the Lobachevsky plane, Theory of Prob.Appl., 3, 391–398.Gruet J C. (1996), Semi-groupe du mouvement brownien hyperbolique, Sto-chastics and Stochastics Reports, 56, 53-61.Gruet J C. (2000), A note on hyperbolic von Mises distribution, Bernoulli, 6,1007-1020.Orsingher E., Beghin, L. (2004), Time-fractional telegraph equations andtelegraph processes with Brownian time, Probab. Theory Related Fields, 128, N.1,141–160.
Universita di Roma “La Sapienza”p.le A.Moro 5, 00185 Rome, Italye-mail: [email protected]
206
Accuracy of approximation of stationaryrandom processes
Palamarchuk Oleksandra Yevgenivna, Ukraine
Let X(t), t ∈ T, T = [0, 1] be a stationary strictly sub-Gaussian randomprocess.
Denote by S := tn, n = 0, N − 1 = n/N, n = 0, N − 1 the parti-tion of the segment [0, 1] into N equal parts. We approximate the randomprocess X(t), t ∈ T by an interpolation broken line XN (t) for given valuesX(tn), n = 0, N − 1, i.e.
XN (t) = α1X(tn) + α2X(tn+1), t ∈ [tn, tn+1], n = 0, N − 1,
where α1 = 1− (t− tn)N, α2 = (t− tn)N.
The aim is to construct a process XN (t), which approximates the processX(t) with given reliability and accuracy in norm of Banach space C([0, 1]) . Soit should be found such least number N that having divided the interval [0, 1]into N equal parts and knowing values of given process in corresponding pointsk/N, k = 0, N, we could restore the process X(t), t ∈ T by the broken lineXN (t), t ∈ T with given accuracy ε and reliability 1 − δ. It means that thenext inequality holds:
P
„supt∈T
|X(t)−XN (t)| > ε
«≤ δ.
Denote by YN (t) := X(t) − XN (t), t ∈ T, the deviation random process.Remark that the process YN (t), t ∈ T is also strictly sub-Gaussian.
Assume that for given process X(t), t ∈ T the next inequality is satisfied:
supt∈T
E|X(t+ h)−X(t)|2 ≤ b2(h), (1)
where b(h), h > 0 is a monotonically increasing continuous function and b(h) ↓ 0as h ↓ 0. Function b(h) for the given random process X(t), t ∈ T is assumedto be known.
Let‘s consider the next two examples for the function b(h).
Example 1. Power function b(h). Assume that the function b(h) for given processX(t), t ∈ [0, 1] have the following form:
b(h) = chα, α ≤ 1, c = const.
Let c = 1, α = 1.
207
Theorem 1 Suppose that for a strictly sub-Gaussian random processX(t), t ∈ T the inequality supt∈T E|X(t+ h)−X(t)|2 ≤ b2(h), whereb(h) = h, holds. If for u ≤ 1
N3ε2the condition u ≤
√17
2(e1/2−1)is satisfied, then
Psupt∈T
|YN (t)| ≥ ε ≤ 2e · exp
−ε
2N2
2
ff“ε2
4√
17√
2 ·N5/2 + 1”4
. (2)
Let accuracy ε = 0.1 and reliability 1− δ = 0.9. Then, applying inequality 2from the theorem, we obtain: N ≥ 83. So the reconstruction of the given processX(t), t ∈ T with accuracy ε = 0.1 and reliability 1− δ = 0.9 is the broken lineXN (t), where N ≥ 83.
Setting higher accuracy ε = 0.01 and reliability 1 − δ = 0.97, we receivethe next result: N ≥ 902, i.e. the desired approximation is the interpolation lineXN (t), where N ≥ 902.
Example 2. Logarithmic function b(h). Assume that
b(h) =c
(ln(1 + dh
))η, η > 1/2, c, d = const.
Let‘s consider the case when c = 1, d = 1.
Theorem 2 If for a strictly sub-Gaussian random process X(t), t ∈ T theinequality 1 is satisfied for b(h) = 1
(ln(1+ 1h
))η , then for ε > A2η
2η+1 ε1
2η+1 · 2η+1
(2η)2η
2η+1,
where A =√
2 · 614η (ln(1+N))
1− 12η
1− 12η
, the next estimate holds
P
supt∈T
|YN (t)| ≥ ε
ff≤ 2 exp
(− 1
2ε20
ε−A
2η2η+1 ε
12η+1 · 2η + 1
(2η)2η
2η+1
!2)(3)
Let η = 4.Setting ε = 0.1 and 1−δ = 0.9 and applying the inequality 3 of the last theo-
rem we obtain N ≥ 14. It means that the desired approximation for given randomprocess X(t), t ∈ T with given accuracy ε = 0.1 and reliability 1 − δ = 0.9 isa broken line XN (t), where N ≥ 14.
Consider higher accuracy and reliability. Let now ε = 0.01 and 1− δ = 0.97.Then we get that the required reconstruction of given strictly sub-Gaussian ran-dom process X(t), t ∈ T with accuracy ε = 0.01 and reliability 1− δ = 0.97 isa broken line XN (t), where N ≥ 171.
It should be noted that a sub-Gaussian random process is a particular caseof ϕ -sub-Gaussian random process with ϕ(x) = x2/2. The obtained results caneasily be generalized to a class of ϕ -sub-Gaussian random processes.
Kyiv National Taras Shevchenko Universitye-mail: [email protected]
208
Asymptotic properties of supervised Bayesianclassifiers in a high-dimensional framework
Tatjana Pavlenko, Sweden
Introduction. Modern experimental techniques make it possible to collect andanalyse huge amounts of data which are mapped to a very high dimensional vari-able space. This data usually carries information about functionalities of certainobservations and are used to infer valuable knowledge for the purpose of classi-fication and prediction. In this study, we focus on the high-dimensional statis-tical model for supervised classification where the covariance structure of class-conditional distributions is sparse. This means that only a few underlying featurevariables, or feature subset are strongly associated with a class variable and ac-count for nearly all of its variation - that is, determine the class membership.Sparse structure is particularly relevant in modeling associations in very highdimensional data and makes it possible to relax the "curse of dimensionality"problem in statistical estimation.
Main results of this paper concern asymptotic properties of the supervisedclassifiers in a high dimensional framework: we show that for a sparse data struc-ture, a supervised classifier turns out to be a special case of generalized additivemodels (see [1]), and its asymptotic distribution approaches normal one underrather mild regularity conditions imposed on the class conditional probabilities.Using asymptotic normality of the classifier we establish a closed form expressionfor asymptotic Bayes misclassification risk which is a natural measure of clas-sification accuracy. Furthermore, we present a feature selection technique whereevaluating the discriminative power of a subset is controlled by the informationabout the response variable and thereby reveal the subsets that are of specialinterest for class separation.Main results. We focus on a supervised classification model, given a training setof continuous feature variables x = (x1, . . . , xp) as well as their associated classmembership variable Y = 1, . . . , C . The task is to build a rule for assessing theclass membership of an observed vector x0 . Let the class conditional probabilitydensities at feature vector x be f(x, θj) , where θj specifies a parametric modelFΘj , j = 1, . . . , C . According to Bayes decision theory, we use the following rule
Y = j if Pr(Y = j|x) = maxk
Pr(Y = k|x), (1)
where Pr(Y = j|x) ∝ πjf(x; θj) is a discriminant score that indicates to whichthe degree x belongs to the class j . Pr(Y = j) = πj are class prior probabilities,j = 1, . . . , C and ∝ denotes proportionality.
This approach can be combined with plug-in estimation of f(x, θj) and isstraightforward in the cases when the number of observations is greater than thatof the dimensionality, i.e. when n > p and p remains fixed. However, it becomesa serious challenge with very high dimensional data, where n p and a plug-inclassifier will be far from optimal.
To overcome this problem we introduce a merging transform:M : Rp×1 → Rq×1 such that (x1, . . . ,xq) with q p are independent
209
functional feature subsets, which constitute a disjoint and complete partitionof the initial set of variables: ∪qi=1xi ⊆ x1, . . . , xp and xi ∪ xj = ∅ forany i 6= j . M is constructed as the supervised feature merging method (see[2]and [3]), which identifies the desirable subsets assuming that a subset is highlydiscriminative if it is tightly clustered given Y , but well separated from the othersubsets.
By applying M to the initial feature set we show that the classifier (1) al-lows factorization Pr(Y = j|x) ∝ πj
Qqi=1 f
ji (xi; θ) , and using plug-in estimates
f ji (xi; θ) we get an additive Bayesian classifier of the form
GM(x; θj , θk) =
qXi=1
GMi (xi; θji , θ
2k) =
qXi=1
lnfi(xi; θ
ji )
fi(xi; θki )(2)
where estimates θi satisfy the standard set of "good" properties such as as-ymptotic unbiasedness and efficiency, uniformly in i as n → ∞ , i = 1, . . . , q ,j, k = 1, . . . , C .
We investigate asymptotic properties of (2) in growing dimension framework,i.e. assuming that limn→∞ p/n = c , where 0 < c < ∞ and show that given thedimensionality of xi fixed to mi (mi n), so that q →∞ , (2) can be consideredas a sum of growing number of independent random variables and its distributionis asymptotically normal under proper regularity conditions imposed on the familyFM,Θ . Asymptotic normality of the classifier (2) gives a closed form expressionfor asymptotic Bayesian misclassification risk
p limn→∞
RGM(x;θj ,θk) = Φ“−√D(j,k)
2
1q1 + 2mρ
D(j,k)
”, (3)
where
D(j,k)i = E
hlnf(xi; θ
ji )
f(xi; θki )
˛f(x; θj)
i− E
hlnf(xi; θ
ji )
f(xi; θki )
˛f(x; θk)
i,
Φ(x) = 1√2π
R x−∞ exp(−z2/2)dz , D(j,k) =
Pqi=1D
(j,k)i is the total cross-entropy
distance between classes j and k (see [4]) and ρ = limn→∞qn
.Asymptotic result (3) gives us means to introduce a feature selection technique
where the relevance of ith subset of features for preserving class separability canbe estimated by its input towards the total cross-entropy distance D(j,k) .
1. Hastie T, Tibshirani R, Friedman, J: The Elements of Statistical Learning: Data Min-ing, Inference and Prediction, Springer, 2001.2. Liu H, Yu L: Towards integrating feature selection algorithms for classification andclustering. IEEE trans knowl and data eng 2005, 3:1-11.3. Pavlenko T, von Rosen D: On the optimal weighting of high-dimensional Bayesiannetworks. Adv and appl in stat 2004, 4:357-377.4. Wang X, Zidek J: Deviation of mixture distributions and weighted likelihood functionas minimizers of KL-divergence subject to constraints. Ann Inst Statist Math 2005, 57:687-701.
Department of Technology, Fysics & Mathematics, Mid Sweden University,851 70 Sundsvall, Sweden; e-mail: [email protected]
210
Whittaker type sampling restoration ofRuscheweyh derivatives for stochastic
processesTibor K. Pogany, Croatia
For f entire having spectral type ψf < qπ2
we have
f(z) = σq(z)X
(m,n)∈Z2
q−1Xj=0
q−1−jXk=0
f (q−1−j−k)(m+ in)
j!(q − 1− j − k)!·
Rjqmn(z −m− in))k+1
,
with
Rjqmn = limw→m+in
dj
dwj
„w −m− ni
σ(w)
«q,
uniformly on all compact z –subsets of C , (Pogany, 2001;2003). Here σ(z) denotesthe Weierstraß σ–function. (The case q = 0 was proved by J.M.Whittaker in1935).
Let (Ω,F,P) be a probability space and denote Lα(Ω,F,P) =: Lα(Ω) spaceof complex r.v.s with the quasi-norm (E| · |α)1/α := ‖ · ‖α, α ∈ (0, 2] en-dowed and on L0(Ω) the topology is the one induced by convergence in prob-ability. Let ξ(t)| t ∈ C be a Lα(Ω)–stochastic process. Then there ex-ists a probability space (eΩ, eF, eP) with L2(Ω) ⊂ L2(eΩ) , a Karhunen processη(t)| t ∈ C ⊂ L2(eΩ) and a random variable Ξ ∈ L2α/(2−α)(Ω) such thatξ(t) = ΞPη(t), t ∈ C , where P is the orthogonal projection from L2(eΩ) toL2(Ω) . Denote L2(Fη; Λ) := spx|
RΛ|x|2dFη <∞ . The derivatives of ξ we take
in the α–mean.Define the Ruscheweyh derivation operator of the order (p, q), p ∈ N0, q ∈ N
with respect to the σ -function as
D(p,q)σ [g]z =
(−1)pσp+q(z)
p!
∂p
∂zp
»g(z)
σq(z)
–and assume f ∈ Leont′ev[2, πψ
2] ;ψ < qθ−(1−θ)p, θ ∈ (0, 1) , where Leont′ev[a, b]
denotes the class of entire functions of order at most a , and when it is equal tob , the type has to be less then or equal to b . Then the Whittaker type q orderderivative sampling restoration formula of D(p,q)
σ [f ]z , reads as followsD(p,q)σ [f ]z =
=X
(m,n)∈Z2
q−1Xl=0
q−1−lXk=0
`p+kk
´σp+q(z)f (l)(m+ ni) lim
ζ→m+ni
∂q−1−l−k
∂ζq−1−l−k
“ζ−m−niσ(ζ)
”ql!(q − 1− l − k)! (z −m− ni)p+1+k
.
The convergence is uniform on all compact z –subsets of C .We write for circular–truncated Whittaker type derivative sampling sum
Ir(t; D(p,q)σ [ξ]) =
X(m,n):m2+n2<r2
q−1Xl=0
q−1−lXk=0
`p+kk
´σp+q(t)f (l)(m+ ni)Rq−1−l−k
qmn
l!(q − 1− l − k)! (t−m− ni)p+1+k.
211
Let ξ(t) ∈ Lα(Ω) be spectrally represented in the formξ(t) =
RΛf(t, λ)dZξ(λ), t ∈ C where supp(Zξ) = Λ ∈ B(C) . Here the
entire function f(t, λ) satisfies
|f(t, λ)| ≤ eLf (λ)(1 + |t|µ)ec?(λ)|=t|
for certain fixed µ ∈ N, supλ∈ΛeLf (λ) := Lf <∞ and ψ = supλ∈Λ c
?(λ) is finiteand has been described already (Piranashvili, 1967). Such a process will be calleda Piranashvili α–process in the sequel.Theorem 1. Let ξ(t) be Piranashvili α -process with p α -mean derivatives, i.e.
∂j
∂tjf(t, λ) ∈ L2(Fη; Λ), j = 0, p.
Then
τr(D(p,q)σ [ξ]; t;α) = ‖D(p,q)
σ [ξ]t − Ir(t; D(p,q)σ [ξ])‖αα
≤
Lf‖Ξ‖2α/(2−α)
pBη
(√
2 K1)q
!α r(1 + |t|m) e−π[qr2−(p+q)|t|2−ψr]/2
(r − |t|)p+1Hq(r)
!α,
where
K1 =“
1− π4
360
”“1− π2
24
“G − π2
15
””2
e−π4 .
Here G =Pn∈N(−1)n−1(2n− 1)−2 is Catalan’s constant and
H(r) :=minr2 − [r2], [r2] + 1− r2
2r + 1/√
2(r ≥ 0) .
When r →∞ we get
D(p,q)σ [ξ]t =
XZ2
q−1Xl=0
q−1−lXk=0
`p+kk
´σp+q(z)f (l)(m+ ni) lim
ζ→m+ni
∂q−1−l−k
∂ζq−1−l−k
“ζ−m−niσ(ζ)
”ql!(q − 1− l − k)! (z −m− ni)p+1+k
in the α -mean and a.s. P , for all q > p .
1. Z.A. Piranashvili, On the problem of interpolation of random processes,Teor. Ver. Primenen. XII/4, 708-717, 1967. (in Russian)
2. T. Pogany, Derivative uniform sampling via Weierstraß σ(z) . Truncationerror analysis in [2, πq/(2s2)) , Georgian Math. J. 8(2001), 129-134.
3. T. Pogany, Local growth of the Weierstraß σ - function and Whittaker -type derivative sampling, Georgian Math. J. 10(2003), 157-164.
4. J.M. Whittaker, Interpolatory Function Theory, Cambridge UniversityPress, Cambridge, 1935.
Department of Sciences, Faculty of Maritime Studies,University of Rijeka, 51000 Rijeka, Studentska 2e-mail: [email protected]
212
Full adaptive wavelet-based estimator ofmixture component density
Dmytro Pokhylko, Ukraine
We consider the problem of density estimation in the case when the observedsample is taken from the mixture with varying concentrations. I.e. we suppose thatthe sample ξjNj=1 consists of the independent random variables, whose distrib-utions have densities, and pj(·) (the density of ξj ) satisfies following statement:
pj(x) =
MXm=1
wmj pm(x), (1)
where M is the number of components in the mixture, pm is the density of them -th component, wmj is the concentration. The concentrations are supposed tobe known. Our aim is to estimate the component’s density.
We will use the projection onto wavelet bases to construct the estimatorpm. Let W = (wmj )Nj=1,
Mm=1 be a matrix with M columns and N rows and
em = (0, . . . , 0, 1, 0, . . . , 0)T the m -th basis vector in RM . Define
am = WΓ−1em, (2)where Γ = 1
NWTW, see [1] for details. Let φ, ψ be the father and mother wavelet,
φjk(x) := 2j/2φ(2jx− k), ψjk(x) := 2j/2ψ(2jx− k). Denote
αmjk =1
N
NXi=1
φjk(ξi)ami , β
mjk =
1
N
NXi=1
ψjk(ξi)ami , P rjpm =
Xk∈Z
αmj0kφj0k(x), (3)
βmjk = βmjk1I|βmjk| > tj, pm(x) =Xk∈Z
αmj0kφj0k(x) +
j1Xj=j0
Xk∈Z
βmjkψjk(x). (4)
where ami is the i -th coordinate of vector am, defined in (2). Let η > 0 be somereal number, denote
Cj,η = maxl
‚‚‚P rjpl‚‚‚+ 2η, K(Cj,η, γ,m) =
= γ
0@1 +4
3‖ψ‖∞ max
k|amk | ln 2 +
vuut(4
3‖ψ‖∞ max
k|amk | ln 2)2 +
8
γCj,η
1
N
NXi=1
(ami )2
1A .
Denote:
ε = sp− p′ − p
2, s′ = s− 1/p+ 1/p′, α =
s/(1 + 2s), ε ≥ 0s′/(1 + 2s− 2/p), ε ≤ 0
,
and Bspq(M,T ) = f ∈ Bspq, ‖f‖spq ≤ M,diam supp ≤ T, f − density, whereBspq is a Besov space equipped with norm ‖·‖spq .Theorem 1 Assume that supN maxi,l |ali:N | < ∞, p′ ≥ p ≥ 1, s − 1
p> 0,
C∞ := supm ‖pm‖∞ < ∞. Suppose φ, ψ ∈ Cr to be father and mother wavelets
213
with compact supports, pm ∈ Bspq,m = 1,M . If tj = K(Cj′,η, γ,m)
qjN,
where η > 0, γ > γ0 = max(( α1−2α
− s′), 1)p′, j′ : 2j′' N
28(s−1/p)+5 , and
2j0 ' (N(lnN)p′−p
p1Iε≥0
)1−2α, 2j1 ' (N(lnN)−1Iε≤0)α/s′, there exist con-
stants C = C(s, p, q,M ′, C∞) and N0 > 0 such that for N > N0 :
suppm∈Bspq(M,T )
E‖pm − pm‖p′
p′ ≤
≤ C( maxi=1,N
|ami:N |)p′
8><>:(lnN)(1−ε/sp)αp
′N−αp′ , ε > 0
(lnN)(p′/2−p/q)+(lnN/N)αp
′, ε = 0
(lnN/N)αp′, ε < 0
,
where x+ = max(x, 0) and pm defined in (3)-(4).Theorem 2 Fix an integer r0 > 0, real µ > 0 and define a class
Sµ := (s, p, q) : 1/p+ µ ≤ s < r0, 1 ≤ p ≤ p′, 1 ≤ q ≤ ∞.
Assume that supN maxi,l |ali:N | + 1 < ∞, p′ ≥ p ≥ 1, s − 1p
> 0,C∞ := supm ‖pm‖∞ < ∞. Suppose φ, ψ ∈ Cr0 to be father and mother waveletswith compact supports, densities pm,m = 1,M belong to some class Bspq(M,T ),where (s, p, q) ∈ Sµ.
If tj = K(Cj′,η, γ0,m)
qjN, where η > 0 , γ0 = r0p
′ + 1, 2j′' N
28µ+5 , and
2j0 ' N1/(1+2r0), 2j1 ' N/ lnN, there exists constants C = C(s, p, q,M,C∞),N0 > 0 such that for N ≥ N0 :
suppm∈Bspq(M,T )
E‖pm − pm‖p′
p′ ≤
≤ C( maxi=1,N
|ami:N |)p′(
(lnN)(p′/2−p/q)+( lnN
N)αp
′, ε = 0
(lnN/N)αp′, ε 6= 0
,
where x+ = max(x, 0) and pm defined in (3)-(4).Remark The case of homogenious observation was considered in [2]. Thresholdedestimator constructed in [2] has near-optimal convergence rate (up to a logarith-mic factor) but requires the apriory information about C∞ to be constructed. Herewe considered the estimator with the same rate of convergence which can be con-structed without information about C∞ .
[1] R.E.Maiboroda, Estimation of components distributions for mixtures with varyingconcentrations, Ukrainian J. Math. 48(1996), no. 4. pp.562-566.
[2] Donoho D., Johnstone I., Kerkyacharian G, Picard D, Density estimation by wavelettresholding, Annals of Statistics, 24(1996) p.508-539.
National Taras Shevchenko University of Kyiv,Faculty of Mechanics and Mathematics,Department of Probability Theory and Math. StatisticsVolodymyrska 64, 01033, Kyiv, Ukrainee-mail: [email protected]
214
Goodness of–fit–test in multivariateerrors-in-variables model
Polekha Maria Yaroslavivna, Ukraine
Consider the model of observations
bi = b0i + bi, ai = a0i + ai, b
0i = XT a0
i , i ≥ 1. (1)
Here a0i ∈ Rn×1, b0i ∈ Rp×1 are unknown nonrandom vectors, the sequences
ai, i ≥ 1 and bi, i ≥ 1 are two IID centered sequences of random errors,independent of each other, and X ∈ Rn×p is a parameter of interest. We assumethat cova =: Sa is known, while covb =: Sb is unknown. Hereafter a d
= ai, bd= bi.
For the model (1) we construct a goodness – of – fit test (GOFT) based on anadjusted least squares (ALS) estimator. The test is propagation of a GOFT fora polynomial measurement error model [1]. Suppose that ai, bi, i = 1,m areobserved. The ALS estimator of X is
X := H−1abT ,
here H := aaT −EaaT , and bar stands for averaging, e.g., abT := 1m
mPi=1
aibTi .
Lemma 1. Assume that the following conditions.(i) E‖a‖4 <∞, E‖b‖4 <∞.
(ii) There exists V∞ := limm→∞
a0a0T , and V∞ is positive definite.
Then H is nonsingular with probability tending to 1,bX P−→ X, and bSb := bbT − baT bX P−→ Sb, as m→∞.
We test a hypothesis H0 about the validity of the model (1). Introduce a teststatistics based on residuals,
T 0m :=
1
m
mXi=1
(bi − XT s(ai))eλT ai ,
where s(ai) := ai−E(aeλT a)/E(eλ
T a) and λ := (λ1, λ2, ..., λn)T is fixed, λk 6= 0,k = 1, n.
Lemma 2.Assume (i), (ii) and the following.(iii) ∃δ > 0 : E(1 + ‖a‖2+δ)e(4+δ)λ
T a <∞.
215
(iv) ∃M((a0(j))l(a0(k))reλT a0), l, r ≥ 0, l + r ≤ 3, and
M((a0(j))l(a0(k))re2λT a0), l, r ≥ 0, l + r ≤ 2; j, k = 1, n.
Hereafter M(f(a0)) stands for a finite limit of averaging:
M(f(a0)) := limm→∞
1m
mPi=1
f(a0i ),
and a0(j), a0(k) stand for j-th and k-th coordinates of a0 ∈ Rn×1.
(v) ∃δ > 0 : ‖a0‖4+δ + e(2+δ)λT a0 + ‖a0‖2+δeλT a0 ≤ const.
Then√m · T 0
md−→ N(0,ΣT ), where
ΣT := Sb ·M [E(eλT a − aT f)2] +XT [1, fT ] ·M(U) · [1, fT ]TX,
f := V −1M(a0eλT a0)E(eλ
T a),
M(U) := limm→∞
cov(Z(a)),
Z(a) :=
"(a0 − s(a))eλ
T a
vec(H)− vec(a0aT )
#, i = 1,m.
Under the conditions of Lemma 2, a consistent estimator Σ of ΣT is con-structed. The test is then defined as follows:
T 2m := m‖Σ−1/2T 0
m‖2.
Theorem 1 Assume the conditions of Lemma 2. Also assume that at least oneof the following two conditions is satisfied:
(vi) M [E(eλT a − aT f)2] > 0, and Sb is positive definite.
(vii) n ≥ p, rankX = p and M(U) is nonsingular.Then T 2
md−→ χ2
p, under hypothesis H0.
We show that under certain local alternative, the proposed test has a noncentralchi-squared asymptotic distribution with p levels of freedom.
References
[1] Cheng C.-L., Kukush A. A goodness-of-fit test in a polinomial errors-in-variables model. Ukrainian Mathematical Jurnal, 2004, 56, 4, 527-543.
Kyiv National Taras Shevchenko Universitye-mail: [email protected]
216
Symbolic-numeric algorithms for quantitativeanalysis of stochastic systems with delay
Igor E. Poloskov, Russia
An analysis of random processes in nonlinear dynamic systems of differenttypes is a very important subject both for theory and practice. The necessity ofsuch analysis is urgent for study of various phenomena: (i) a flight of vehicles underan action of atmospheric turbulence; (ii) a traffic along a rough road; (iii) high-altitude vibrations of structures under wind and seismic attacks; (iv) a rolling ofships due to a rough sea etc.
When we solve a significant number of practical problems and use the theoryof Markov processes, we can assume that a random vector process x ∈ Rn , aphase vector, which describes a status of an object being studied, satisfies a setof stochastic differential equations (SDEs) in the Stratonovich sense [1]
x(t) = f(x, t) +G(x, t)ξ(t), x(t0) = x0, (1)
where ξ ∈ Rm is a vector of independent Gaussian white noises with unit inten-sities, f(·, ·) = fi(·, ·)T : Rn × [t0,∞) → Rn is a deterministic vector-function,G(·, ·) = gij(·, ·) : Rn × [t0,∞) → Rn × Rm is a deterministic matrix-function,and T is a symbol of the transposition.
The main stochastic characteristics of x are the probability density function(PDF) p(x, t) , the transition probability density function (TPDF) p(x, t|y, τ) ,the moments mα = M[xα] and µα = M[(x−M[x])α] , where α = α1, α2, ..., αnis a multi-index (αi ≥ 0), and cumulants. Among moments we can high-light the first ones such as the mean values mi = M[xi] = 〈xi〉 , the vari-ances Di = σ2
i = 〈(xi − mi)2〉 , the mixed moments mij = 〈xixj〉 and
Dij = 〈`xi −mi
´`xj −mj
´〉 of the second order. Here Dij are elements of the
matrix D of covariances and Mˆ·˜
stands for the mathematical expectation.It is well known that the TPDF p(x, t|y, s) satisfies the Fokker-Planck-
Kolmogorov (FPK) equation
∂p
∂t= −
nXi=1
∂`ai p´
∂xi+
1
2
nXi,j=1
∂2`bij p
´∂xi∂xj
, limt→s+0
p(x, t|y, s) = δ(x− y), (2)
ai(x, t) = fi +1
2
nXj=1
mXk=1
∂gik∂xj
gjk, B(x, t) ≡ bij(x, t) = G(x, t)GT (x, t). (3)
The PDF p(x, t) satisfies equation (Eq.) (2) too but under the initial conditionp(x, t0) = p0(x) .
But there are many stochastic problems, which can’t be described by Eqs.(1).Here we consider one of such problems concerned with a necessity of taking intoaccount of an aftereffect in systems under a research.
217
We study the following set of nonlinear stochastic differential-difference equa-tions for a non-Markovian random vector process x(t)
x(t) = fν(x(t),xτ (t),x2τ (t), ...,xντ (t), t)+Gν(x(t),xτ (t),x2τ (t), ...,xντ (t), t) ξ(t),
where τ is a constant delay, t > tν = t0 + ν τ , xkτ (t) = x(t − kτ) , ν > 0 isan integer. It is supposed that on the segments (t0, t1] , (t1, t2] , ..., (tν−1, tν ] , thephase vector x(t) satisfies the following systems of SDEs
x(t) = f0(x(t), t) +G0(x(t), t)ξ(t),
x(t) = f1(x(t),xτ (t), t) +G1(x(t),xτ (t), t) ξ(t),
.....................................................
x(t) = fν−1(x(t),xτ (t),x2τ (t), ...,x(ν−1)τ (t), t)+
+Gν−1(x(t),xτ (t),x2τ (t), ...,x(ν−1)τ (t), t) ξ(t).
To obtain the PDF p(x, t) at any t > t0 , we expand the phase space of thesystem and reduce a non-Markovian vector process to a Markovian one by thisway. Using the following notation
s ∈ [0, τ ], tq = t0 + q · τ, q = 0, 1, 2, ..., xq(s) = x(s+ tq), ξq(s) = ξ(s+ tq),
pq(xq, s) = p(x, s+ tq), z0 = x0, z1 = col(x0,x1), z2 = col(x0,x1,x2), ...,
p0(x0, 0) = p0(x0), yq ≡ xq(0) = xq−1(τ), ξq(0) = ξq−1(τ), pq(xq, 0) = pq−1(xq, τ),
we construct a chain of FPK-like equations for the TPDFs of the vectorsz0 , z1 , z2 , ..., zN , ... belonging to the family of embedded phase spacesRn ⊂ R2n ⊂ R3n ⊂ ... ⊂ Rn(N+1) ⊂ ... .
Then we derive chains of equations for the PDFs, equations for the first mo-ments for linear systems with additive noises, and for nonlinear systems on thebase of the Gaussian approximation. Further we define a number of stochasticcases for a classic method of steps and show how to exploit the method of mo-ments and the cumulant closure at a stage of numerical calculations.
The technique is applied to study a number of linear and nonlinear models,linear equations of car motion effected by front-to-rear delay and a rough road,dynamics of pollutions discharged into a cascade of natural water bodies. Thepackage Mathematica [2] is a tool of such studies.
References
[1] Dimentberg, M. F. (1980). Nonlinear Stochastic Problems of Mechanical Vi-brations. Nauka. Moscow (in Russian).
[2] Wolfram, S. (1999). The Mathematica Book, 4th edn. University Press. Cam-bridge.
Perm State University,POB 1652, Perm-39, 614039, Russiae-mail: [email protected]
218
Random Linear Operators in Banach Spaces
Oleksandr Ponomarenko, Ukraine
Random linear operators acting in linear topological spaces are very compli-cated objects for study. The systematic theory of such objects now exists only inthe case of random linear operators acting in one separable Hilbert space. Suchtheory was constructed by A.V.Skorohod in the book "Random linear operators",Kiev, 1978. In this communication we consider the general probabilistic charac-teristics of different types of random linear operators which act from one Banachspace X to another separable Banach space Y .
Let L(X,Y ) be a space of all continuous linear operators acting from X toY and (Ω,F ,P) be some probability space. Denote by Y (Ω) the class of all Y -valued random elements endowed by topology of convergence in probability andby Y (Ω) the class generalized random elements on Ω in Y , i.e. class of continuouslinear mapping L(Y ∗,R(Ω)) , where Y ∗ is a dual space for Y [2]. We study thefollowing classes of random operators.
(i) The class L(Ω;X,Y ) of all bounded random linear operators A(w), w ∈ Ω ,which act from X to Y , where A(w) are weakly measurable L(X,Y ) -valuedrandom elements.
(ii)The class Ls(Ω;X,Y ) of all strong random linear operators A(w), w ∈ Ω ,which act from X to Y , where A(w) ∈ L(X,Y (Ω)) , i.e. A(w) continuously andlinearly maps elements x ∈ X into random elements A(w), x ∈ Y (Ω) .
(iii)The class Lw(Ω;X,Y ) of all weak random operators A(w), w ∈ Ω ,which act from X to Y , where A(w) ∈ L(X, Y (Ω)) , i.e.for x ∈ X(A(w)(x)) is generalized random elements in Y . There are the strong inclusionsLw(Ω;X,Y ) ⊃ Ls(Ω;X,Y ) ⊃ L(Ω;X,Y ) . We study the additional conditionsunder which random operator from wider class belongs to more narrow class.
The conceptions of conjugate random operator and product of operators fromdifferent classes are discussed. The notation of characteristic functions of randomoperators is introduced and properties of such functions are investigated. The re-sults allow us to give definitions of Gaussian random operators, their mean andcovariance, operator-valued Brownian motion. Then we consider general class ofweak operator of order p ≥ 1 and define moments characteristic of such operators.The main attention is devoted to the class of second order weak random opera-tors. The notion of conditional expectation of random operator is introduced. Atlast we study weak and strong convergence of a sequence of weak and strongrandom operators respectively and weak convergence of distributions for randomoperators.
Kiev Taras Shevchenko National University,Faculty of Mechanics and Mathematicse-mail: [email protected]
219
Spectral analysis of some classes ofmultivariate random fields with isotropic
property
Oleksandr Ponomarenko, Yuriy Perun, Ukraine
Let X be a complex normed space with dual space X∗ and second dual spaceX∗∗ and H = L2(Ω) be Hilbert space of all second-order random variables onprobability space (Ω,F ,P) . The space L(X∗, H) of linear continuous mappingΞ : X∗ → H can be considered as space of generalized secon-order random ele-ments in X with mean (EΞ) ∈ X∗∗ and covariance [Ξ,Ψ] ∈ AL(X) (space ofcontinuous antilinear operators C : X∗ → X∗∗ ) uniquely defined by relations(EΞ)(u) = E(Ξu) , ([Ξ,Ψ]ν)(u) = E(Ξu)(Ψν), u, ν ∈ X∗,Ξ,Ψ ∈ L(X∗, H) [1].
Let Ξt, t ∈ Rn be a generalized random field in X which is continu-ous in strong topology of L(X∗, H) , has zero mean and covariance functionB(t, s) = [Ξt,Ξs] . The field Ξt is homogeneous if B(t, s) = B(t− s, 0) = K(t− s)for all t, s ∈ Rn , is exponentially convex if B(t, s) = B(t + s, 0) = K(t + s) forall t, s ∈ Rn and and is isotropic if B(gt, gs) = B(t, s) for all t, s ∈ Rn and allg ∈ SO(n) (group of rotations in Rn around 0).
Theorem 1 Let Ξt, t ∈ Rn be an exponentially convex random field in X . Thenthere exist such L(X∗, H) -valued random orthogonal Radon measure Φ on Rnand AL(X) -valued Radon measure F on Rn , [Φ(∆1),Φ(∆2)] = F (∆1 ∩ ∆2) ,that take place the following representations
Ξt =
ZRn
exp(λ, t)Φ(dλ), K(t) =
ZRn
exp(λ, t)F (dλ),
where (·, ·) is scalar product in Rn .
This theorem contains as partial case the results of M.Loeve about one-dimensional exponentially convex stochastic processes [2].
Theorem 2 If Ξt, t ∈ Rn is an exponentially convex and isotropic random fieldin X , then the following representations are valid
K(t) =
Z ∞
0
Yn(iλ‖t‖)G(dλ),
Ξt = cn
∞Xm=0
h(m,n)Xl=1
Slm(θ1, ..., θn−2, ϕ)
Z ∞
0
(iλ‖t‖)2−n
2 Jm+ n−2
2(iλ‖t‖)Φlm(dλ),
(1)where Jr is Bessel function of first type and order r , (‖t‖, θ1, ..., θn−2, ϕ) arespherical coordinates of vector t, Slm are orthogonal spherical harmonics of degree
220
m and h(m,n) is their number, G is positive AL(X) -valued Radon measure onR+ = [0,∞) , Φlm are L(X∗, H) -valued Radon measure on R+ with
[Φlm(∆1),Φqp(∆2)] = δmpδlqG(∆1 ∩∆2)
cn = 2n−1Γ(n
2)π
n2 , Yn(z) = 2
n−22 Γ(
n
2)z
2−n2 Jn−2
2(z).
Note that first representation in (1) is operator analogue of Nussbaum theoremabout complex -valued exponentially convex radial function [3].
Theorem 3 If Ξt, t ∈ Rn is homogeneous and isotropic random field in X , thenthe spectral representations of the form
K(t) =
Z ∞
0
Yn(λ‖t‖)G(dλ),
Ξt = cn
∞Xm=0
h(m,n)Xl=1
Slm(θ1, ..., θn−2, ϕ)
Z ∞
0
(λ‖t‖)2−n
2 Jm+ n−2
2(λ‖t‖)Φlm(dλ),
take place with notations introduced in the theorem 2 .
The results of this theorem are operator version of Shoenberg theorem [4]about integral representation of positive definite radial function and Yadrenkotheorem [5] about spectral representation of one-dimensional homogeneous andisotropic random field.
References
1. Ponomarenko O.I. Second-order Random Linear Functionals. I., TheoryProb. Math. Statist. 54 (1997), p.143-152; Second-order Random LinearFunctionals. II., Theory Prob. Math. Statist. 55 (1997), p.165-175.
2. Lo eve M. Fonctions al eatoires a charact ere exponentiel, In P. Levy, Proces-sus stochastiques et mouvement Brownien, Paris, Gauthiers-Villars, 1965.
3. Nussbaum A. E. Radial exponentially convex functions, J. Anal. Math., v.25,(1972), p.277-288.
4. Shoenberg I.J.,Metric space and completely monotone functions., J. Anal.Math., v.39, 1938, p.811-841.
5. Yadrenko M.I., Spectral theory of random fields., Kiev, Kiev Univ. Press,1980 (in Russian).
Kiev Taras Shevchenko National University,Faculty of Mechanics and Mathematicse-mail: [email protected]
221
Integrated representation of the semigroup ofoperators which describes the diffusion processwith piecewise-continuous matrix of diffusionand generalized transfer vector concentrated
on a surfaceMykola Portenko, Bohdan Kopytko, Ukraine
Let in Euclidian space Rn , n > 2 , some closed hypersurface S be definedthat subdivides Rn to two open sets: internal D1 and external Dn so thatRn = D1 ∪ D2 ∪ S . We assume that S is a surface of the Holder class H1+λ
for some λ ∈ (0, 1) (see [1]).Suppose that in D1 , D2 the forming differential operators of some diffusion
processes L1 and L2 are defined:
Ll =1
2
nXi,j=1
b(l)ij Dij +
nXi=1
a(l)i Di, l = 1, 2,
where Di = ∂/∂xi , Dij = ∂2/∂xi∂xj , i, j = 1, . . . , n ; b(l)ij (x) and a
(l)i (x)
are bounded continuous functions in Rn , b(l)ij ∈ Hλ(Rn) , a
(l)i ∈ Hλ(Rn) ,
i, j = 1, . . . , n , l = 1, 2 , and the matrices B(l)(x) =“b(l)ij (x)
”are symmetric,
positive and continuously non-singular.The problem is to construct the semigroup of operators Tt , t > 0 , that de-
scribes the Feller process in Rn such that in the parts Dl , l = 1, 2 , it coincideswith the diffusion processes which are formed by Ll , l = 1, 2 . The domain of thecharacteristic operator of the process in the class of smooth functions at the pointsof S is determined by means of a first order linear differential operator in spa-tial variables that contains only the derivatives in the direction of the co-normalvectors.
To solve the problem we will use analytic methods. Due to the above mentionedapproach the required semigroup and its process is determined by the solution ofthe correspondent conjunction problem of the second degree linear parabolic equa-tion with discontinuous coefficients. The classical solution of the above mentionedproblem is obtained by the boundary integrated equations method. Moreover, weprove that the constructed process may be considered as generalized diffusionprocess (see [1]).
[1] M. I. Portenko, The generalized diffusion processes. Naukova Dumka, Kiev,1982.
Institute of Mathematics, National Academy of Sciences of Ukraine,Kiev, 01601, Ukraine; e-mail: [email protected]
Department of Mathematics, Ivan Franko Lviv National University,Universytetska 1, Lviv, 79000, Ukraine; e-mail: [email protected]
222
Existence and uniqueness of solution ofstochastic differential equation, driven by
fractional Brownian motion
Sergey Posashkov, Ukraine
The stochastic differential equation written in integral form
Xt = X0+
tZ0
a(s,Xs)ds+
tZ0
b(s,Xs)dWs+
tZ0
c(s,Xs)dBHs +ε
tZ0
c(s,Xs)dVs, t ∈ [0, T ],
(1)is considered. Coefficients a, b, c : [0, T ]× R → R are measurable functions, V,Ware independent Brownian motions, ε > 0 , and BH is independent fractionalBrownian motion with Hurst parameter H ∈ (3/4, 1] .
Theorem 1 Suppose that the coefficients of equation (1) satisfy the followingconditions:1. For all r > 0 there exists lr such that
(a(s, x)− a(s, y))2 + (b(s, x)− b(s, y))2 + (c(s, x)− c(s, y))2 6 lr(x− y)2,
for |x| 6 r , |y| 6 r , s 6 r .2. There exists L such that
(a(s, x))2 + (b(s, x))2 + (c(s, x))2 6 L(1 + x2).
Then there exists F ′t -measurable solution of equation (1), whereFt = σX0,Ws, εVs +BHs , s ∈ [0, t], and this solution is unique.
1. P. Cheredito, Regularizing Fractional Brownian Motion with a View towardsStock Price Modelling, ”Swiss Federal Institute of Technology”, Zurich, 2001.
2. M. Hitsuda, Representation of Gaussian processes equivalent to Wienerprocess, Osaka Journal of Mathematics 5 (1968), 299-312.
3. И. И. Гихман, А. В. Скороход, Стохастические дифференциальныеуравнения и их приложения, ”Наукова думка”, Киев, 1982.
Kyiv National University,Chair of Theory of Probability and Mathematical Statistics,Department of Mathematics and Mechanics,Kyiv National University, Glushkov pr-t 6, Kyiv, Ukrainee-mail: [email protected]
223
On Lebesgue measure of sets of numbersdefined by the properties of their Ostrogradsky
seriesMykola Pratsiovytyi, Oleksandr Baranovskyi, Ukraine
Any real number x ∈ (0, 1) can be represented by finite or infinite expression(the first Ostrogradsky series) [1]
1
q1− 1
q1q2+ · · ·+ (−1)n−1
q1q2 . . . qn+ · · · = O1(q1, q2, . . . , qn, . . . ), (1)
where qn ∈ N and qn+1 > qn for all n . One can rewrite series (1) in the form
1
g1− 1
g1(g1 + g2)+ · · ·+ (−1)n−1
g1(g1 + g2) . . . (g1 + g2 + · · ·+ gn)+ · · · . (2)
We denote the expression (2) briefly by O1(g1, g2, . . . , gn, . . . ) . A representationof a number x ∈ (0, 1) by expression (2) is called the O1 -representation. Thenumbers gn are called the O1 -symbols of the number x .
We calculated Lebesgue measure of subsets from some classes of closednowhere dense sets defined by characteristic properties of the O1 -representation.
Let C[O1, V ] consists of all numbers x ∈ [0, 1] , whose O1 -symbols qk aresuch that qk ∈ V , where V is a fixed subset of N . We obtained the conditionsfor the set C[O1, V ] to be of zero resp. positive Lebesgue measure.
Let us consider the following random variable
ξ = O1(ξ1, ξ2, . . . , ξk, . . . ),
where ξk are independent random variables taking the values 1 , 2 , . . . ,m , . . . with probabilities p1k , p2k , . . . , pmk , . . . correspondingly, pmk ≥ 0 ,∞Pm=1
pmk = 1 .
We prove that the random variable ξ is of pure type, i.e., it is either purecontinuous or pure atomic. In the atomic case we completely describe the set ofall atoms of the distribution. In the case of singularity the conditions for suchdistributions to be of Cantor type are also obtained.
References
[1] E. Ya. Remez, On series with alternating sign which may be connected with twoalgorithms of M. V. Ostrogradskiı for the approximation of irrational numbers,Uspehi Matem. Nauk (N.S.) 6 (1951), no. 5 (45), 33–42 (Russian).
Institute of Mathematics, National Academy of Sciences of Ukraine,Tereshchenkivska str. 3, 01601 Kyiv-4, Ukraine;Dragomanov National Pedagogical University,Pyrogova str. 9, 01601 Kyiv, Ukrainee-mail: [email protected]
224
Regularity of generated by stochastic measuresgeneralized random functions
Vadym Radchenko, Ukraine
Let (X, B) be a measurable space and L0 be a set of all real random variableswith topology of convergence in probability. Let µ be a stochastic measure (ingeneral sense) on B , i. e. µ be a σ -additive function µ : B → L0 .
The integration of real functions with respect to stochastic measures is consid-ered in [1, 2]. All bounded measurable functions are integrable w.r.t. any stochasticmeasure. The dominated convergence theorem for this integral holds true.
By B (R) denote the Borel σ -algebra of R . Let D = D (R) be the set ofC∞ functions of compact support in R . For any defined on B (R) stochasticmeasure µ we define the generalized random function µ by
(µ, ϕ) =
ZR
ϕdµ, ϕ ∈ D.
Theorem 1 For any stochastic measure µ defined on B (R) generalized randomfunction µ has a regular modification (i. e. a version with values in D′ ).
Another regularity result deals with continuity of paths of parameter stochasticintegrals.
Let T be a metric space, function f : T×X → R is integrable w.r.t. µ on Xfor any fixed t ∈ T . Consider random function
η(t) =
ZX
f(t, x) dµ(x), t ∈ T . (1)
Theorem 2 Let X = [a, b] ⊂ R , B be the Borel σ -algebra of [a, b] , T be ametric space. Suppose for each x ∈ [a, b] f (·, x) is continuous on T and forsome γ > 1/2, C > 0 for all t ∈ T, x, y ∈ [a, b]
|f(t, x)− f(t, y)| ≤ C |x− y|γ . (2)
Then η(t) has a version with continuous on T paths.
References
[1] S. Kwapien, W.A. Woyczinski Random series and stochastic integrals: singleand multiple. Boston: Birkhauser, 1992.
[2] V. N. Radchenko. Integrals of general stochastic measures. Kyiv: Institute ofMathematics of NAS of Ukraine, 1999 (Russian).
Department of Mathematical Analysis,Kyiv National Taras Shevchenko University, Kyiv 01033 Ukrainee-mail: [email protected]
225
On the estimate for the covariance function ofhomogeneous and isotropic vector-valued random field
on the sphere
G.M.Rakhimov, D.G.Rakhimova, Uzbekistan
Let ξ(P, t) : P ∈ Sn a homogeneous on time variable and isotropic on a spacevariable vector-valued random field with components ξa(P, t)(a = 1, . . . , r) onSn ×R , where Sn is the unit sphere Sn of the n -rd dimensional space Rn , andR = (−∞,+∞) which has finite second order moment and which for each t iscontinuous in quadratic mean (q.m.). Eξ(P, t) = µ, µ− unknown.
Let field ξ(P, t) is observed for all points of Sn and on time [0, T ] . As anestimate of the covariance function of a random field ξ(P, t) the following statisticsis considered:
BT (θ, t) = ω−1n
NTXm=0
h(m,n)C
n−22
m (cos θ)
Cn−2
2m (1)
rTm(t)
where
a,brTm(t) =
+∞Z−∞
eiλt a,bITm(λ) dλ
a,bITm(λ) is the periodogram, 0 ≤ θ ≤ π , h(m,n) = (2m + n − 2) (m+n−3)!
(n−2)!m!
is the number linearly independent spherical harmonics of degree m ,ωn = 2πn/2
‹Γ(n/2)− of the area of the surface Sn and Cνm(ν 6= 0) - Gegen-
bauer polynomials, for each T,NT is a positive integer and NT →∞ as T →∞ .In this paper we study asymptotic distribution estimate of the covariance functionBT (θ, t) of a random fields ξ(P, t) .
Institute of Mathematics, Institute of Law, Uzbekistane-mail: [email protected]
226
Superstability, Strong Superstability andConstruction of Gibbs measure.
Oleksiy Rebenko, Sergiy Petrenko, Ukraine
We consider infinite system of interacting particles in Rd . The phase space isthe configuration space Γ which is the set of all locally finite sets of Rd . Denote byΓ0 the set of all finite configuration. Let ∆ is a partition of Rd into nonintersectingd -cubes with a rib λ .
Definition 1The system of interacting particles is called superstable (p = 2) orstrong superstable (p > 2) if there exists constants A > 0 and B ≥ 0 such thatfor any η ∈ Γ0 the potential energy of the system satisfies the following enequality:
U(η) ≥X∆∈∆
(A|η∆|p −B|η∆|),
where η∆ is the projection of a configuration η on ∆ .
The main statement of the report is the following
Lemma 1 Let U(η) =Px,y⊂η φ(|x − y|) and φ(|x|) ≥ ϕ0/|x|α at |x| ≤ λ ,
ϕ0 > 0 , then there exist constant aα > 0 such that
U(η∆) =
8<:aα|η∆|2, for α < d,aα|η∆|2 ln |η∆|, for α = d,aα|η∆|p, p > 2, for α > d,
Remark 1 In the case α < d the system is superstable at
aα > supx1∈Rd
X∆∈∆
supx2∈∆
φ−(|x1 − x2|), φ−(|x|) = min0, φ(|x|).
In the case α = d the system is superstable and for α > d the system is strongsuperstable.
As an example we apply these criterions for the construction of Gibbs measurein the problems which are connected with diffusion dynamics of infinite particlesystems in Rd .
Institute of Mathematics, Kieve-mail: [email protected]
227
The problem of the choice of the investmentprojects portfolio under stochastic uncertainty
Roskladka Andriy, Ukraine
The problem of a choice of a portfolio of investment projects is considered.During the investment period η packages of the spare capital for purchase of blockof shares k of the enterprises is expected. The set of packages of the spare capitalcan be presented as set of n groups on ηj packages in cost ej .
The investment politics imposes limiting restrictions: V1 - the minimal volumeof means for investment; V2 - the minimal volume of means which are necessaryfor holding in the highly liquid form; V3 - the maximal volume of means whichare put in shares which rate has significant fluctuations.
Let researches of an infrastructure of the market have allowed to take intoaccount uncertainty of volumes of capital investments and to establish thus guar-anteed probabilities of fulfillment of three limiting restrictions concerning randomvariables V1, V2, V3 which above mentioned.
Let’s designate through xj cost of the got shares of the enterprise j , andthrough cj - expected profit on a unit of cost of the invested shares which, obvi-ously, also is a random variable.
The given problem has the combinatorial nature, because in her the choiceby the investor of some set of share holdings from the fixed amount packagesof shares of the enterprises is carried out. If to consider multiset G with a ba-sis S (G) = (e1, e2, ..., en) and the primary specification [G] = (η1, η2, ..., ηn) ,where (η1 + η2 + ...+ ηn = η) , corresponding sets of packages of shares are nat-ural for considering as elements of the general Euclidean combinatorial set [1]of arrangements Ekηn (G) , which elements are ordered k - samples of multiset Gwhich contains η elements among which is n different.
The mathematical model of a problem of the choice of investment projectswith the purpose of maximization of the expected profit can be represented insuch a way:
M
kXj=1
cjxj
!→ max (1)
under the conditions
P
kXj=1
dijxj ≤ bi
!≥ αi, (2)
(x1, x2, ..., xk) ∈ Ekηn (G) . (3)
From the economic contents of a problem comes up, that value of factors dijsuch: d1j = −1, j = 1, 2, 3; d2j = −1 , if shares of the enterprise j are highlyliquid shares and d2j = 0 differently; d3j = 1 , if the share price of the enterprisehas significant fluctuations and d3j = 0 differently.
The right parts of restrictions (3) such: b1 = −V1 , b2 = −V2 , b3 = −V3 .
228
For a finding of the solution of this stochastic problem we shall pass to deter-mined equivalent of target function (1):
kXj=1
cjxj → max, (4)
where cj = M (cj) , and to the determined equivalents of restrictions (2):kXj=1
dijxj ≤ bi, (5)
in which parameters bi are the roots of the equation 1 − Fi“bi”
= αi , whereFi (bi) - function of distribution of casual parameter bi [2].
Thus the stochastic problem (1) - (3) is reduced to the determined problemof linear programming (4), (5) with additional combinatorial restriction (3).
The problem (1) - (3) is a special case of the general task of Euclidean com-binatorial optimization under uncertainties [3, 4]:
S F (〈x〉 , ω) → extr, (6)
ϕi (〈x〉 , ω) ≤ 0 ∀i ∈ Jm, 〈x〉 ∈ eE, ω ∈ Ω, (7)where 〈x〉 = (〈x, νx〉 |µx) – fuzzy element of interval space I (Rn) , given with anerror of a measurement νx and with a value function of a membership, equal µx ;m – integer non-negative constant; ω – casual parameter, which characterizes adefined condition of an environment; Ω – set of these conditions; F (〈x〉 , ω)– atarget functional with an interval value, which depends also from ω ; S – somestatistical function of a stochastic content; ϕi , i ∈ Jm – some, generally speak-ing, stochastic functions; Jm – set of the first m natural numbers; eE – the fuzzyEuclidean combinatorial set with by random properties from the elements of in-terval space: eE ⊂ I (Rn) . Each of the elements of a model (6) - (7) is consideredas depending from a series of parameters[5].
1. Стоян Ю.Г., Ємець О. О. Теорiя i методи евклiдової комбiнаторної оптимiзацiї.– К.: Iнститут системних дослiджень освiти, 1993. –188 с.
2. Юдин Д. Б. Задачи и методы стохастического программирования. – М.: Сов.Радио, 1979.– 392 с.
3. Roskladka A. Stochastic settings of the problems of Euclidean combinatorial opti-mization // Theory of stochastic processes,9 (25), no.3-4, 2003. - P. 170-175.
4. Yemets O. O., Roskladka A. A. About the general approach of modelling by thetasks of optimization on Euclidean combinatorial sets under uncertainties // AbstractsInternational Workshop Prediction And Decision Making Under Uncertainties (PDMU-2005 ) (September, 12-17, 2005, Berdyansk). - P. 72-73.
5. Роскладка А. А. Параметричнi задачi та стiйкiсть при моделюваннiевклiдовими комбiнаторними задачами оптимiзацiї: Дис. ... канд. фiз.-мат. наук:01.05.01. – Полтава: ПДТУ, 2000. – 142 с.
Полтавський унiверситет споживчої кооперацiї України,кафедра математичного моделювання та соцiальної iнформатикиe-mail: [email protected]
229
Multifractional Markov processes
Maria D. Ruiz-Medina, Spain; Vo V. Anh, Australia; Jose M. Angulo, Spain
We introduce a class of Markov processes whose transition probability densi-ties are defined by multifractional pseudodifferential evolution equations on com-pact domains with variable local dimension. The infinitesimal generators of theseMarkov processes are given by the trace on domains with variable local dimensionof strongly elliptic pseudodifferential operators of variable order. The results de-rived provide a pseudomultifractal version of some existing special classes of mul-tifractional Markov processes. In particular, pseudostable processes are definedon domains with variable local dimension in this framework. The local Holderexponent of their transition probability densities is obtained. In the case wherethe local dimension of the domain and the local Holder exponent of the transitionprobability densities are constant, the existing results on fractal versions of Levyprocesses are recovered.
[M.D. Ruiz-Medina and J.M. Angulo]Department of Statistics and Operations ResearchUniversity of GranadaCampus Fuente Nueva s/n, E-18071 Granada, Spain
[V.V. Anh]School of Mathematical SciencesQueensland University of TechnologyGPO Box 2434, Brisbane Q 4001, Australia
Estimation of mixing probabilities by censoreddata
Anton Ryzhov, Ukraine
The talk deals with a problem of estimation of unknown mixing probabilitiesin the case of right censored data. Consider two populations X1 and X2 withcumulative distribution functions (CDF) of failure H1(t) and H2(t) , t ≥ 0 re-spectively. The observable data consists of N > 2 samples such that for everyi = 1, . . . , N CDF of the i’th sample is Fi(t) = θiH1(t) + (1 − θi)H2(t), t ≥ 0with unknown weight 0 ≤ θi ≤ 1 .
The case when θ1 ≡ 1 , θN ≡ 0 is considered. We propose estimators ofunknown values of θi, i = 2, . . . , N−1 based on Kaplan-Meier [1] estimators Fi(t)of CDFs Fi(t), i = 1, . . . , N and generalized least squares method (GLSM) [2]. Itis shown, that estimator θn of the array θ = (1, θ2, . . . , θN−1) can be obtained asa solution to minimization problem
αT Bnα
αTSnα→ min
230
on the set Θ = 1 × [0, 1]N−2 of all α = (1, α2, . . . , αN−1) . Here Bn = (bnlm)and Sn = (snlm) are two matrices of size (N − 1) × (N − 1) with elements
bnlm =PN−1i,j=1
ninj
n2
τR0
Fi(t)Fj(t)`1l=i − nl
n
´ `1m=j − nm
n
´λ(dt),
and snlm = nln
`1l=m − nm
n
´respectively, nl denotes size of the l’th sample,
n = n1+· · ·+nN – total size of all samples, λ(·) – Lebesgue measure on [0, τ ] . Thechoice of end-point τ depends on properties of underlined CDFs F1(t), . . . , FN (t)and CDFs of censoring C1(t), . . . , CN (t) .
Theorem 1(Strong consistency of θn ) Assume that for every j = 1, . . . , N thereexist constant 0 < hj < 1 such that
limn→∞
njn
= hj
and ∃t ∈ [0, τ ] : H1(t) 6= H2(t) . Then θn is strongly consistent, that is,ρ(θn, θ) → 0 a.s. as n→∞, where ρ – Euclidian distance on Θ .
Theorem 2(Asymptotic normality of θn ). Assume that for all j = 1, . . . , NCDFs Fj(t) and Cj(t), t ≥ 0 are continuous on [0, τ ] . Under assumptions ofTheorem 1 √
n(θn − θ) =⇒ ζ(θ), n→∞and limit vector ζ(θ) = (ζ2(θ), . . . , ζN−1(θ))T has multidimensional Gaussian dis-tribution with mean 0 and covariance matrix Ξ(θ) .
An exact expression for matrix Ξ(θ) is derived.
References
[1] Kaplan E.L., Meier P., Nonparametric estimation from incomplete observa-tions, J. Am. Statist. Assoc. 53 (1958), 457–481.
[2] Maiboroda R.E., Statistical analysis of mixtures, VPC “Kyivskiy universytet”,Kyiv, 2003 (in Ukrainian).
Dept. of Mechanics and Mathematics, Kyiv University,Vladimirska 64, Kyiv, Ukrainee-mail: [email protected]
231
Comparison of Quasi Score and CorrectedScore Estimators in Multivariate Nonlinear
Measurement Error Model
Anastasiia Rzhevska, Ukraine
Let Y be a random vector distributed in Rk , which has a probability densityfunction belonging to the exponential family:
f(Y |ξ) =
„Y Tξ − C(ξ)
ϕ+ c1(Y, ϕ)
«, (1)
where a column vector ξ ∈ Rk is the canonical parameter and ϕ > 0 is a dispersionparameter, which may be either known or unknown. I assume that ξ is known.The function C : Rk −→ R is supposed to be sufficiently smooth. The parameterξ is a function of some random vector X ∈ Rn and an unknown parameter vectorβ ∈ Θβ ⊂ Rm (Θβ is a compact set): ξ = ξ(X,β).
The density of Y , as given above, is therefore a conditional density given X .Note that the conditional mean of Y given X is given by:
E(Y |X) = C′ (ξ(X,β)) (2)
and the conditional covariance matrix equals
Var(Y |X) = ϕC′′ (ξ(X,β)) . (3)
We assume that the matrix C′′ is positive definite. The parameter of interestis β . In a measurement error model, X is unobservable. Instead we observe thesurrogate vector variable W , which is related to X as
W = X + U,
U being the unobservable measurement error. The random vector U is sup-posed to be independent of (X,Y ) . The problem is to estimate β from asample (Yi,Wi), i = 1, . . . p , of observable data. We assume that the triples(Yi, Xi,Wi), i = 1, . . . p are independent.
There exist several consistent estimators of β if additional information on thedistribution of X and U is available.
We introduce two unbiased score functions:
SQ = mTβ v
−1(Y −m) Quasi Score (QS), (4)
SC = gTY − hT Corrected Score (CS), (5)
wherem := m(W,β) := E(Y |W ) ∈ Rk, (6)
232
v := v(W,β, ϕ) := E“
(Y −m)(Y −m)T|W”∈ Rk×k (7)
and g = g(W,β), h = h(W,β) functions which are the solutions to the deconvolu-tion problems:
E(g|X) = ξβ ∈ Rk×m, (8)
E(h|X) = (C′)Tξβ ∈ Rm. (9)
Note that the derivative of a column vector valued function f(β) with respect toβ is always meant to be a matrix fβ = ∂f
∂βT with (i, j) -element ∂fi∂βj
.
Also, we consider the so-called Simple Score (SS):
SS = gT(Y −m). (10)
The scores SQ , SC and SS are unbiased, which means that at true pointESQ = ESC = ESS = 0.
We prove that the QS estimator of β is more efficient then the CS estimatorof β in the sense that asymptotic covariance matrix (ACM) of the QS estimatoris less or equal (in the Loewner sense) to the ACM of CS estimator. SS estimatorserves as an intermediate estimator in comparing the efficiency of QS and CS.
We also give conditions under which the strict inequality holds true.Some examples of exponential family models are presented. Namely, we con-
sider Gaussian regression multivariate model and Loglinear Poisson multivariatemodel under assumption(N) :X and U are independent random variables and X ∼ N(µx, σ
2x), U ∼ N(0, σ2
u).
We show how CS and QS estimators can be constructed and give conditionsfor a strict inequalities between ACM of QS and CS.
In addition we prove that ACM of β does not change in the case where ϕ isunknown and has to be estimated with the help of sone score functions like, e.g.S(ϕ) = Y TY −mTm− tr(v) or S(ϕ) = (Y −m)T(Y −m)− tr(v).
The results generalize the results from Kukush et al.(2005) where a univariatemodel (1) was studied, with k = 1.
References
A.Kukush, H.Schneeweiss and S.Shklyar, Quasi Score is more efficient than Cor-rected Score in a general nonlinear measurement error model, Discussion Paper451, SFB 386, LMU, Munich, 2005.
Kiev National Taras Shevchenko Universitye-mail: [email protected]
233
Asymptotic properties of spectral estimatesLudmila Sakhno, Ukraine
We consider empirical spectral functionals of second and higher orders anddiscuss the conditions which guarantee the convergence in probability of thesefunctionals, and also allow to evaluate their moments and asymptotic distribution.Results on asymptotic properties of the empirical spectral functionals can be usedfor parametric and non-parametric estimation of random processes and fields withshort- or long-range dependence. Some examples of applications are provided.
The talk is based on the results obtained in collaboration with N.N. Leonenkoand V.V. Anh.
References
[1] V.V. Anh, N.N. Leonenko, and L.M. Sakhno, Quasilikelihood-based higher-orderspectral estimation of random processes and fields with possible long-range depen-dence, J. Appl. Probab. 41A, 2004, 35-54.
[2] V.V. Anh, N.N. Leonenko, and L.M. Sakhno, Minimum contrast estimation of ran-dom processes based on information of second and third orders, J. Stat. Plan. Infer.,2006, to appear.
Kyiv National Taras Shevchenko Universitye-mail: [email protected]
Spectral properties of scaling limit solutions ofpartial differential equations with random
initial conditionsLudmila Sakhno, Ukraine
We present the higher-order spectral densities of random fields arising as ap-proximations of rescaled solutions of the heat, fractional heat, Burgers and KPZequations with long-range dependent initial conditions.
The talk is based on the results obtained in collaboration with N.N. Leonenkoand V.V. Anh.
References
[1] V.V. Anh, N.N. Leonenko, and L.M. Sakhno, Higher-Order Spectral Densities ofFractional Random Fields, J. Statist. Phys. 111, 2003, 789-814.
[2] V.V. Anh, N.N. Leonenko, and L.M. Sakhno, Spectral Properties of Burgers andKPZ Turbulence, J. Statist. Phys., 2006, to appear.
Kyiv National Taras Shevchenko Universitye-mail: [email protected]
234
Asymptotic expansion for Markov randomevolutions
Samoilenko Igor, Ukraine
We study Markov random evolution
Φεt (u, x) = E[ϕ(uε(t),æεt )|uε(0) = u,æ(0) = x], (1)
where x ∈ E , u ∈ Rd , æ(t) - Markov process with intensity matrix Q ,æεt := æ(t/ε).
The Markov random evolution has a generator V(x)ϕ(u) = V (u, x)ϕ′(u) , forthe function uε(t) we may write the following stochastic system
duε(t) = V (uε(t),æεt )dt.
We are looking for asymptotic expansion of (1) in the view
Φεt (u, x) = U0(t) +
∞Xk=1
εk(Uk(t) +W k(t)). (2)
Theorem. Regular terms of the expansion (2) have the view: U0(t) = c0(t)1,where c0(t) satisfies the equation
ΠV(x)Π∂c0∂u
− ∂c0∂t
= 0.
The regular terms are the following: Uk(t) = R0Φ(Uk−1(t)) + ck(t), whereΦ(Un(t)) = dUn
dt− V(x)Un(t), scalar functions ck(t) satisfy the equations
ΠV(x)Π∂ck∂u
− ∂ck∂t
=
kXn=1
(−1)nCknΠV(x)R0Vn(x)Πdn−1
dtn−1c0.
Boundary layer is the following: W (1)(x, t) = exp0(Qt)W 1(x, 0), fork > 1 : W (k)(x, t) = exp0(Qt)W (k)(x, 0) +
R t0exp0(Q(t − s))V(x)W (k−1)(x, s)ds.
Initial conditions:
c(0)(0) = f(x), U (1)(x, 0) = R0V(x)f(x),W (1)(x, 0) = −1
2V(x)Πf(x),
for k > 1 : W (k)(x, 0) = ΦhUu
(k−1)(x, 0)
i, c(k)(0) = −V(x)fW (k−1)(x, 0), wherefW (1)(x, 0) = −R0V(x)f(x),fW (k)(x, 0) = R0Φ
hu(k−1)(x, 0)
i+R0V(x)fW (k−1)(x, 0)
+ΠV(x)(fW (k−1)(x, λ))′λ|λ=0,
(fW (k)(x, λ))′λ|λ=0 = R20Φhu(k−1)(x, 0)
i+R2
0Q1fW (k−1)(x, 0)
+R0V(x)(fW (k−1)(x, λ))′λ|λ=0.
Institute of Mathematics of NASU01601, Tereschenkivska str, 3, Kiev, Ukrainee-mail: [email protected]
235
Small sample modification of adjusted leastsquares in linear errors-in-variables model
Ivan Senko, Ukraine
Let existing such A0 ∈ Rm×n, B0 ∈ Rm×d and X0 ∈ Rn×d thatA0X0 = B0. Matrices A = A0 + A, B = B0 + B are observed whereA = (aij)
m,ni=1,j=1, B = (bik)m,di=1,k=1 are errors. The matrix VA = E AT A sup-
posed to be known.Let’s introduce the further assumptions:(i) aij , i > 1, 1 6 j 6 n and bik, i > 1, 1 6 k 6 d form two independent
arrays of r.v., which are centered and possess finite second order moments.(ii) the rows of A are independent and the rows of B are independent.(iii) supE a4
ij : i > 1, 1 6 j 6 m+ supE b2ik : i > 1, 1 6 k 6 d < +∞(iv)
`λmin
`AT0 A0
´+m
´‹λ2
max
`AT0 A0
´→ 0, m→∞ .
(v) the rows of A are identically distributed and the rows of B are identicallydistributed; E
ˆa11 . . . a1n
˜T ˆa11 . . . a1n
˜6= 0 .
(vi) Covˆb11 . . . b1d
˜T> 0 and ∃ limm→∞
1mAT0 A0 > 0 .
(vii) distributions L(a1j), 1 6 j 6 n; L(b1k), 1 6 k 6 d have no atoms.Adjusted least squares estimator X = (ATA − VA)†ATB of the unknown
value X0 is consistent if assumptions (i)-(iv) hold.Modified adjusted least squares
estimator is built as following. Let T ,
»BTB BTAATB ATA
–; W ,
»0 00 VA
–.
Let λA be the smallest positive root of det (T − λW ) = 0 . We set
µ ,
m−αm
, λA > 1 + 1m,
λAm−αm+1
, λA 6 1 + 1m,
where 0 < α < m fixed arbitrary number. So
XM ,`ATA− µVA
´ATB . Under conditions (i)–(viii) the next statements hold.
Lemma 1∀m > n+ d : T > 0 a.s. So λA exists for m > n+ d
Lemma 2∀m > n + d : ATA − µVA > α+1m+1
ATA > 0 a.s. So XM should havebetter properties for small samples than X .
Theorem 1√m“X − XM
”P→ 0, m → ∞, i.e. XM and X have the same
asymptotic properties.
References1. A.Kukush, I.Markovsky, S. Van Huffel. Consistent estimation in the bilinear
multivariate errors-in-variables model. Metrica(2003) 57:253-2852. C.-L. Cheng, H. Schneeweiss, M. Thamerus A small sample estimator for a
polynomial regression with errors in the variables. J. R. Statist. Soc. B.(2003) 62Part 4. pp. 699-709.
Kiev National University after T.Shevchenkoe-mail: [email protected]
236
Consistent Estimation of RegressionCoefficients in Measurement Error Model
under Exact Linear Restrictions
Shalabh, India
In linear regression models, the ordinary least squares estimator (OLSE) isinconsistent and biased when the observations on variables are observed withmeasurement errors. It is well known that in order to obtain the consistent es-timators of regression coefficients, some additional information from outside thesample, e.g., measurement error variance, ratio of measurement error variances orreliability ratio etc. is required.
Moreover, some prior information about the regression coefficients is some-times available which can be used to improve upon the OLSE. The restrictedleast squares estimator is used when the data is observed without measurementerrors and such prior information can be expressed in the form of exact linear re-strictions binding the regression coefficients. This estimator is consistent, satisfiesthe restrictions and has smaller variability than OLSE.
When the observations are contaminated with measurement errors, the re-stricted least squares estimator becomes inconsistent and biased. So the problemof obtaining a consistent estimator, which also satisfies the restrictions is addressedin this paper.
The multivariate ultrastructural model is considered and no assumption ismade about the distributional form of any of the measurement errors and ran-dom error component in the model. Only the existence and finiteness of first fourmoments of measurement errors and random error component are assumed. Theadditional knowledge of reliability matrix and covariance matrix of measurementerrors associated with explanatory variables is used to obtain the consistent esti-mators which also satisfy the given restrictions.
The bias vectors and mean squared error matrices of the estimators arederived and studied using the large sample approximation theory. An inter-comparison of both the estimators is made and dominance conditions for thesuperiority of one estimator over the other are obtained under structural andfunctional forms of the measurement error models. The effect of departure fromnormal assumption is also studied. A Monte-Carlo simulation experiment is alsoconducted to study the behaviour of estimators in finite samples.
Indian Institute of Technology Kanpur,Department of Mathematics & Statistics,Kanpur - 208016 INDIAe-mail: [email protected]; [email protected]
237
Properties of solutions of stochastic differentialequations in Hilbert spaces
Georgiy Shevchenko, Ukraine
The work is devoted to problems of numerical solution of stochastic differen-tial equations, and to approximation of semigroups of operators. For stochasticdifferential equations in Hilbert spaces numerical solutions are constructed bythe means of Euler’s and Mil’stein’s schemes, finite-dimensional (Galerkin) pro-jections, and regularization of coefficients. Convergence results are proved, andthe rate of convergence is established. The particular emphasis is put on linearequations.
For anticipating stochastic differential equations with Skorohod integral weobtain the results on existence of approximate solutions in spaces of generalizedrandom variables. Discrete-time approximation schemes for numerically solvingquasilinear anticipating equations are constructed. The convergence of approxi-mations is proved, and the rate appears to be the same as for ordinary (non-anticipating) equations.
Keywords: approximation by time discretization, stochastic differential equa-tion, Ito–Volterra equation, stochastic differential equation of evolution type,stochastic differential equation with anticipation, Skorohod integral, white noisespace, semigroup of operators.
References
[1] Shevchenko, G. M. Rate of convergence of discrete approximate solutions forstochastic differential equations in a Hilbert space. Theory Probab. Math. Stat.no. 69 (2004), 187–199.
[2] Mishura Yu. S., Shevchenko, G. M. Linear equations and stochastic exponentsin a Hilbert space. Teor. Imovir. Mat. Stat. no 71 (2004), 123–132.
[3] Mishura, Yu. S., Shevchenko, G. M. Euler approximation of abstract differen-tial equations in Hilbert spaces and their application in semigroup theory, Ukr.Math. Journal 56 (2004), no. 3, 399–410.
[4] Shevchenko, G. M. Approximate integration of stochastic differential equationsDopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, no. 1. (2005), 39–46.
[5] Shevchenko, G. M. On Euler approximations of stochastic differential equationswith anticipation. Teor. Imovir. Mat. Stat. no 72 (2005), 152–159.
Department of Mechanics and Mathematics,Kyiv National Taras Shevchenko University,64 Volodymyrska, 01033 Kyiv, Ukrainee-mail: [email protected]
238
Length of interval of indeterminacy inchange-points problem
Grigorij Shurenkov, UkraineWe consider the problem of search for multiple change-points with sequential
use of dynamic programming approach. We estimated the number of observations,that should be made additionally to ensure that next observations don’t affect theresult obtained at previous steps.
Let ξ1, ..., ξN be a sequence of independent random values, that in generalcase are not identically distributed, but can have one of distributions F1, ...FK .P(ξ ∈ A) = Fh0
j(A), where h0 =
˘h0j , j = 1, ..., N
¯is non-random sequence of
indices h0j ∈ 1, ...,K , h0
j = const when ki = [θiN ] < j ≤ [θi+1N ] , where0 = θ0 < θ1 < ... < θR < θR+1 = 1 are non-random numbers, that are calledmoments of change, ki are called change-points. A sequence of indices of distrib-utions h is called trajectory, h0 is called real trajectory of sequence ξ1, ..., ξN .Suppose for some functions φ(x, j) : R → R, j = 1, ...K the following conditionholds
E mini6=j
(φ(ζi, i)− φ(ζi, j)) > 0,
where ζi has distribution Fi . Let’s introduce the functionalJ t(h) =
Pti=1 (φ(ξi, hi)− πN (hi, hi−1)) , 0 < t ≤ N, πN (g, l) = πN1g 6=l
where πN > 0, is non-random value that we name penalty for change of distrib-ution. Consider the following estimator for h0 h = argmax
hJN (h) . Estimates for
change-points are indices of observations in trajectory h at whose index of distri-bution is changing. Trajectory h can be searched gradually. We should examinegj(l) = argmax Jj(l) for each j , where maximum is taken on trajectories withdistribution l at the moment j . If at some moment j trajectories gj(l) coincidefor l = 1, ...K , until that moment optimal trajectory is already known. Denote τj- index of observation at which trajectories coincide j -th time. Then we shouldestimate the following value ∆τj = τj − τj−1 . It was proven by the author, thatthe following statement holds
τl+1 = min
8<:t > τl|[k
\i6=k
\τl<u≤t
(J t(gt(k))− Ju(gu(i))−
tXj=u+1
φ(ξj , i) > πN
)9=; .
Also under some conditions an estimate for E∆τj was obtained. That estimateshows E∆τj is proportional to penalty for change of distribution.
1. Borovkov A.A. Asymptotically optimal solutions in change-point problem// Theor. of probability and its applications, vol. 43, No. 4, 1998, p.625-654.
2. Sugakova O.V. A search for change points in the flow of independent ob-servations// Theor. Probability and Math. Statist. No.55, 1997, p.181-186.
Kyiv National Universitye-mail: [email protected]
239
Statistical problems for quantization ofrandom processes
Mykola Shykula and Oleg Seleznjev, Sweden
Quantization of a continuous-in-value signal into discrete form (or discretiza-tion of amplitude) is a standard task in all analog/digital devices. Quantizationis a part of digital representation of an analog signal for various conversion tech-niques such as source coding, data compression, archiving, restoration etc. Asymp-totic properties of additive noise quantization models for a wide class of randomprocesses for both uniform and non-uniform scalar quantizers are studied in [4](see also [1] and references therein for comprehensive overview of the problem). Acoding quantization approach is developed in [3]. This approach is similar to run-length encoding (RLE) (see, e.g., [2]). In RLE, a signal that consists of repeatedsymbols is compressed by encoding each repetition as a pair ("symbol", "length ofthe repetition"). This technique is widely used, especially in image compression,where repetition of pixel values are common. We consider average case analysisfor various coding methods and storage techniques of quantized in value/time sig-nals, when a signal is a realization of a random process. For run-length encodingwe predict (in average) the necessary memory capacity needed for a quantizedprocess realization by using the correlation structure of the original process.
In practice, signals are sampled in time. Let X(0), X(h), . . . , X(hn) be an ob-served sample of a stationary Gaussian zero-mean process (signal) X(t), t ∈ [0, T ],with covariance function r(t − s) . Let h = T/n be a sampling step. For apositive ε , consider the uniform quantization with an infinite levels’ ε -gridu(ε) = uk, k ∈ Z := kε, k ∈ Z and with quantizer qε(x) := ε[x/ε] for areal x . Then, the storage memory capacity needed for a quantized process re-alization compressed by RLE (or RLE quantization rate) can be represented asfollows
lε(X) := lε, h(X,T ) =Pn−1j=0 I
`qε(X(hj)) 6= qε(X(hj + h))
´,
where I(·) is the indicator function. Let Lε(X) := Lε, h(X,T ) = E(lε(X)) be themean RLE quantization rate for X(0), X(h), . . . , X(hn) . The goal is to investigatethe asymptotic behavior of Lε(X) as ε→ 0 .
Theorem 1 Let X(0), X(h), . . . , X(hn) be an observed random sample from astationary Gaussian zero-mean process X(t) with covariance function r(t − s),sampling step is h = T/n . Then, for a fixed h > 0 ,
Lε(X) ∼ T
h
„1− 1√
2π
εpr(0)p
r(0)2 − r(h)2
«as ε→ 0.
Thus, the asymptotic mean quantization rate Lε(X) depends on quantizationcellwidth ε , sampling step h , and the correlation structure of the processX(t), t ∈ [0, T ] . The asymptotic estimator bLε(X) of the Lε(X) can be ob-tained using a standard estimator br(h) of r(h) based on the observations
240
X(0), X(h), . . . , X(nh) . This result is generalized for various cases, when ε andh vary in according way. A consistent and asymptotically normal estimate for themean RLE quantization rate Lε(X) is obtained for fixed ε and h , and sufficientlylarge number of observations n .
These results can be useful in communication theory, signal processing, coding,and compression applications. Some examples and numerical experiments demon-strating the rate of convergence in the obtained asymptotic results are discussed.
This is a joint work with Associate Professor Oleg Seleznjev, Department ofMathematics and Mathematical statistics, Umea University, Sweden.
References[1] Gray R.M. and Neuhoff D.L. (1998) Quantization. IEEE Trans. Inform. Theor.44, 2325-2383.[2] Makinen, V. Navarro, G. and Ukkonen, E. (2003). Approximate matching ofrun-length compressed strings. Aigorithmica 35, 347–369.[3] Shykula, M. and Seleznjev, O. (2004). Uniform quantization of randomprocesses. Univ. Umea Research Report, Dep. Math. Stat. 2004:1, 1-16.[4] Shykula, M. and Seleznjev, O. (2006). Stochastic structure of asymptotic quan-tization errors. Stat. Prob. Letters 76, 453–464.
Department of Mathematics and Mathematical Statistics,Umea University, SE-901 87 Umea, Swedene-mail: [email protected]
Consistency of Simex estimators in polynomialerrors-in-variables model
Olena Sidelnyk, Ukraine
A polynomial structural errors-in-variables model is described by the followingequations:
yi = β0 + β1ξi + . . .+ βmξmi + εi,
xi = ξi + δi,i = 1, n.
Assumptions on the model are: ξi, δi, εi are mutually i.i.d sequences,Eξi = µ, E|ξi|m <∞, Eεi = 0, Eε2i <∞, δi ∼ N(0, σ2
δ ) and σ2δ is known.
Let Xi = (xi, . . . , xmi )t . The naive estimator of β is
bβnaive = M−1XXMXY ,
where MXX = 1n
nPi=1
XiXti and MXY = 1
n
nPi=1
Xiyi .
The idea of Simex (Simulation Extrapolation) estimators belongs to Cookand Stefanski[1]. Simex estimates are obtained by adding extra measurement er-ror to the data in a resampling-like stage, establishing a trend of measurement
241
error-induced bias versus the variance of the added measurement error, and ex-trapolation this trend back to the case of no measurement error.
The construction of Simex estimator is performed in two steps: simulationstep and extrapolation step. The bootstrap method on the simulation step isapplied. Let B be the size of bootstrap sample. Generation of standard normalindependent identically distributed variables εib, i = 1, n, b = 1, B is made.For each λ from a set λk, k = 0,K, λ0 = 0 , introduce
MXX(λ) = X(λ)Xt(λ), MXY (λ) = X(λ)y.The overline sign means averaging over all i and b and vector
Xib(λ) =“xi + εib
√λ, . . . , (xi + εib
√λ)m
”t.
Corresponding estimate of β isbβnaive(λ) = M−1XX(λ)MXY (λ).
For construction of the trend model consider the polynomials hi(x, t) which pos-sess the following properties:
h−1(x, t) = h0(x, t) = 1, hi+1(x, t) = xhi(x, t) + it · hi−1(x, t), i ≥ 1.Let H(x,t) be a matrix with entries: Hrs(x, t) = hr+s(x, t), r, s = 0, ...,m , and
T (t) be the transition matrix V (x, t) = T (t)X, where X = (1, x, . . . , xm)t andV (x, t) = (h0(x, t), . . . , hm(x, t))t .
The next parametric model for Simex estimator is proposed:
bβ(λ, θ) =
1
n
nXi=1
H(xi, λ)
!−1
T (λ)θ.
The parameter θ is estimated by the least square method:
bθ = argminθ
KXk=0
||bβnaive(λk)− bβ(θ, λk)||2.
On the extrapolation step we define the Simex estimate as
bβSimex = bβ(−σ2δ , bθ) =
1
n
nXi=1
H(xi,−σ2δ )
!−1
T (−σ2δ )bθ
The consistency of this estimator is proved. In case K=0 the estimate coincideswith the adjusted least estimate [2]
References
[1] Carroll, R.J., Ruppert, D. and Strefanski, L.A. (1996). Measurement error innonlinerar models, London: Chapman and Hall.
[2] Cheng. C.-L. and Schneeweiss, H. (1998).Polynoial regression with errors inthe variables. J.R.Statist.Soc.B, 60, 189-199
Kyiv National Taras Shevchenko Universitye-mail: [email protected]
242
Quasi-Stationary Phenomena in NonlinearlyPerturbed Stochastic Systems
Dmitrii Silvestrov, Sweden
The lecture presents a new book written by me in cooperation with ProfessorMats Gyllenberg (University of Helsinki). The book is devoted to studies of quasi-stationary phenomena in nonlinearly perturbed stochastic systems. The core ofthis phenomenon is that one can observe something that resembles a stationarybehaviour of the system before its lifetime goes to the end. Examples of stochas-tic systems, in which quasi-stationary phenomena can be observed, are variousqueuing systems and reliability models, in which the lifetime is usually consideredto be the time in which some kind of a fatal failure occurs in the system. Pop-ulation dynamics or epidemic models supply another class of examples of suchstochastic systems. In population dynamics models, the lifetimes are usually theextinction times for the corresponding populations. In epidemic models, the roleof the lifetime is played by the time of extinction of the epidemic in the pop-ulation. A ruin for risk processes can be also considered as an example of suchphenomenon. An important characteristic feature of these models is a non-linearcharacter of perturbation that essentially complicates the asymptotic analysis ofquasi-stationary phenomena. The methods developed in the book are based on theexponential asymptotic expansions for nonlinearly perturbed renewal equations.Mixed ergodic and large deviation theorems are given for nonlinearly perturbed re-generative processes, semi-Markov processes, and Markov chains. Applications tononlinearly perturbed population dynamics and epidemic models, queuing systemsand risk processes are considered. The book contains an extended bibliography ofworks in the area.
Malardalen UniversityVasteras, Sweden
243
Necessary and Sufficient Conditions for WeakConvergence of First-Rare-Event Times for
Semi-Markov Processes with Applications toRisk Theory
Dmitrii Silvestrov, Sweden; Myroslav Drozdenko, Sweden
Necessary and sufficient conditions for weak convergence of first-rare-eventtimes for semi-Markov processes with finite set of states are obtained. These re-sults are applied to risk processes and give necessary and sufficient conditionsfor stable approximation of ruin probabilities including the case of diffusion ap-proximation. The most previous results give sufficient conditions of convergencefor such functionals. As a rule, those conditions involve assumptions, which im-ply convergence of distributions for sums of i.i.d. random variables distributedas sojourn times for the semi-Markov process (for every state) to some infinitelydivisible laws plus some ergodicity condition for the imbedded Markov chain pluscondition of vanishing probabilities of occurring rare event during one transitionstep for the semi-Markov process. Our results are related to the model of semi-Markov processes with a finite set of states. In this paper, we consider the caseof stable type asymptotics for distributions of sojourn times. Instead of condi-tions based on “individual” distributions of sojourn times, we use more generaland weaker conditions imposed on distributions (of sojourn times) averaged bystationary distribution of the imbedded Markov chain. Moreover, we show thatthese conditions are not only sufficient but also necessary conditions for the weakconvergence for first-rare-event times, and describe the class of all possible limit-ing laws not-concentrated in zero. The results presented in the papers [1, 2] givesome kind of a “final solution” for limit theorems for first-rare-event times forsemi-Markov process with a finite set of states.
References
[1] Silvestrov D.S., Drozdenko M.O. (2005) Necessary and Sufficient Conditionsfor Weak Convergence of First-Rare-Event Times for Semi-Markov Processeswith Applications to Risk Theory Research Report 2005-2, Department ofMathematics and Physics, Mдlardalen University, 59 pages.
[2] Silvestrov, D.S., Drozdenko, M.O. (2006) Necessary and sufficient conditionsfor weak convergence of first-rare-event times for semi-Markov processes Re-ports Nat. Acad. Sci. Ukraine (to appear)
Department of Mathematics and Physics, Malardalen University,Box 883, 721 23 Vasteras, Swedene-mail: [email protected]; [email protected]
244
Reinsurance AnalyserDmitrii Silvestrov, Sweden
Jef Teugels, BelgiumViktoriya Masol, Ukraine
Anatoliy Malyarenko, Sweden
We introduce a Monte Carlo based approach to evaluate and/or compare theriskiness of reinsurance treaties. Due to the Monte Carlo technique the implemen-tation of the indicated approach extends to a large set of reinsurance treaties. Inparticular, simulation methods allow to cope with the mathematical complexityof extreme value reinsurance contracts.
An experimental program system Reinsurance Analyser based on the indicatedapproach is also presented. Reinsurance Analyser becomes an especially flexibleand handy tool when one compares the riskiness of reinsurance contracts in theinteractive regime. Moreover, the use of computer programs allows us to analyzeapplications of a reinsurance contract under a variety of claim flow models.
The effect of applications of reinsurance contracts is compared by a set of riskmeasures (e.g., value at risk, coefficient of variation, etc.). The key idea is to useMonte Carlo simulations to calibrate the contracts so that their reinsurer’s quotaloads are equal, and to estimate the ratios of the corresponding risk measures ofthe contracts.
Furthermore, we illustrate how one can use Reinsurance Analyser to performexperimental studies on the comparison of the riskiness of reinsurance contracts.We present results of such experiments obtained for an extreme value reinsurancecontract and classical excess-of-loss in case when the variance of the underlyingclaim flow distribution is infinite.
References
1. Teugels, J.L. (2003) Reinsurance. Actuarial Aspects. EURANDOM Report2003-006, Technical University of Eindhoven, The Nederlands.
2. Silvestrov, D.S., Teugels, J.L., Masol, V., Malyarenko, A. (2005) Reinsur-ance Analyser. Research Report 2005-04, Malardalen University, Sweden.
Malardalen University, Box 883, SE-721 23 Vasteras, Swedene-mail: [email protected]
Katholieke Universiteit Leuven, B-3001 Leuven (Heverlee), Belgiume-mail: [email protected]
Kyiv National Taras Shevchenko University, 64 Volodymyrska, Kyiv, Ukrainee-mail: [email protected]
Malardalen University, Box 883, SE-721 23 Vasteras Swedene-mail: [email protected]
245
Teaching Undergraduate and GraduateStatistics Courses: The Importance of
Research ProjectEugenia Skirta, USA
Statistics is a powerful science of working with data and learning from data. Welive in a digital world, and technology allows us to explore statistical ideas in newand exciting ways, particularly by using such professional statistical packages asSAS, SPSS, Minitab, or Excel, and other digital tools. But incorporating a researchproject into undergraduate statistics syllabi, defining a manageable project, andmarketing the project to undergraduates requires several changes. The definitionof statistical thinking presented in the GAISE (Guidelines for Assessment andInstruction in Statistics Education) Report for the college introductory course,endorsed by the American Statistical Association, is “understanding the basislanguage of statistics and understanding some fundamental ideas of statistics.” [1]Among these fundamental ideas are the need to work with data, the importance ofdata production, collection, and use of statistical graphs, and the ability to selectan appropriate technique to answer a research question [2, 3]. I firmly believe thattoday everyone from students and teachers to workers and employers has to beable to interpret data. I have taught introductory and general courses in statisticsalong with advanced statistical courses for graduate students at both, the M Scand PhD levels, for the past sixteen years witnessing the welcome evolution fromthe traditional formula-driven mathematical statistics course to a concept-drivenapproach. This concept-driven approach emphasizes why statistics is importantin the real world, especially these days. Conceptual understanding develops astudent’s ability to make decisions and think critically as well as provides studentswith a strong-problem solving mindset that will help them to succeed in futurestatistics courses, and later, in future professional activities.
This presentation focuses on how to incorporate a service learning componentinto introductory statistics courses as well as how to inspire effective research forgraduate students. To incorporate the project into the introductory or general sta-tistics course, several changes were made. The major shift has been in the deliveryof the course content. Making the material relevant and interesting to the studentsby using current data on the web or using student-generated data is a key elementin the restructuring of the course. I also provide my students with examples ofworksheets and software guides to facilitate their technology-learning process (ei-ther Excel or SPSS), which is an important element for being able to work withdata effectively. Besides it, establishing a collaborative culture is perhaps the mostimportant prerequisite to creating an environment in which students are willingto participate actively in their own learning. While there are many learning goalsof general statistics courses, the most difficult are those that focus on the learningof fundamental concepts, such as variation, prediction, confidence, and hypothesistesting. How can we best measure whether students are able to apply, interpretand communicate these concepts? While many alternative means of assessmenthave been suggested, requiring the students to write about their analysis and con-
246
clusions often reveals hidden inadequacies and misconceptions in learning alongwith a lack of experience of presenting statistical ideas and results.
Undergraduate projects are formulated to require some data collection activ-ity on a relatively small scale. Students see the problems related to working withreal data. They enjoy and appreciate working on research that connects to excit-ing and useful applications. Most of the topics covered in the project are usuallyobtained from government agencies’ web sites, professional publications, and stu-dents’ advisors. In an undergraduate statistics course I help students to identifya broad group of statistical techniques that students have already studied. Stu-dents learn by doing and working in teams. The data analysis part is probablythe easiest part of the project. Students use either Excel or SPSS for the analy-sis. They develop the following skills, among others: collecting their own data;defining the variables that have been measured; entering data into the computer;calculating descriptive statistics; graphical presentations of data; knowing whichstatistical method or test to use; carrying out the test or procedure; interpretingthe statistical output, and presenting the results of the statistical analysis in areport. Undergraduate students really learn by working on a research project, andperhaps most importantly, become more self-motivated, more critical, and morecreative in their approach to material.
Introductory, general statistics, Statistics I and II courses develop the student’sability to apply the statistical knowledge to real-world problems, preparing themto accomplish useful research effectively as undergraduates and to enter graduateresearch programs with their eyes open. In the 2005 fall semester, I taught a grad-uate course “Linear Statistical Modeling Methods with SAS.” The students workedon the midterm research project and completed the course by presenting the finalproject (written and oral reports). They wrote some special purpose programs inSAS for data management activities such as data editing, transformation, pro-ducing various graphs, and data analysis. Since the statistical techniques involvedare covered in the course, the students got farther insight into the subject andgot a real understanding of it. The students developed much advanced statisticalknowledge of multiple regression, stepwise, forward, and backward procedures andother criteria for model selection, interaction regression models, multicollinearitydiagnostics, identifying outlying and influential observations, to name a few. Oralpresentations to teammates during the semester, as well as a presentation to theclass at the end of the semester, provide valuable experience that cannot be taughtby coursework alone.
1.Michael Sullivan, Statistics. Informed Decision Using Data. 2nd Ed, Prentice Hall,NJ, 2006.
2.Gary Smith, Learning Statistics by Doing Statistics, J. of Statistics Education, v.6, n. 3, 1998 (http://www.amstat.org/publications/jse/v6n3/smith.html)
3.Sandra Fillebrown, Using Projects in an Elementary Statistics Coursefor Non-Science Majors, J. of Statistics Education, v. 2, n. 2, 1994(http://www.amstat.org/publications/jse/v2n2/fillebrown.html)
Department of Mathematics, East Stroudsburg UniversityEast Stroudsburg, PA 18301, e-mail: [email protected]
247
Wave Scattering from Composite DielectricAnisotropic Structures
Eugenia Skirta, USA
The effective dielectric permittivity of dielectric anisotropic structures deter-mined by its inclusion’s shape, geometrical size, electromagnetic parameters andspatial packing was first obtained by using a method of averaging of electro-magnetic fields over the volume of a regular bulk dielectric in [1]. The methodpresented in this talk allows us to investigate sufficiently dense media, take intoaccount multiple interactions and the dispersion properties of the media whosescattering centers are either distorted particles of revolution or systems of theregular particles of revolution [2]. In the initial formulation of the problem thereare no restrictions imposed upon the inclusion’s shape and spatial distribution.The method of averaging used in the present paper implies a replacement of thepoint-like scattering center by an origin of scattered waves spread over the “ele-mentary volume” with the spatial density function that is represented by a powerseries in the rectangular coordinates. We computed the mean field with the accu-racy up to the fourth order and obtained the expansion coefficients of the spatialdensity function. The solution yields the mean effective permittivity of a bulk di-electric that is expressed in terms of the geometric and electromagnetic parametersof the structure and particles comprising it. The deformation of dielectric particlesis described by three harmonical functions for symmetrical and nonsymmetricaldeformations. The function of symmetrical deformation, that is a function of thesecond order, determines the deformation of an ellipsoid which has the shape ofellipsoid of revolution uniformly pressed on both butt-ends. The simplest func-tion of nonsymmetrical deformation used in this work is a function of the thirdorder. The latter allows describing the shape of ellipsoid symmetrical about oneaxis only. The last function determines a completely nonsymmetrical deformation,and is also of the third order. The volume of a deformed ellipsoid is equal to thesum of the volume of a regular ellipsoid and the volume of a complementary layerabove and below the surface of the regular particle.
This method enables us to calculate the average dielectric permittivity of ran-domly or regularly spatially distributed scattering elements (point data) of variousshapes and can be used for the purpose of spatially distributed modeling of groundcovers. There are various strategies and mathematical techniques for translatingthe results of the smaller-scale measurements or predictions for applications inlarger-scale modeling. The distributed modeling approach [3, 4] involves physi-cally based point models and attempts to use smaller-scale data into larger-scalemodeling by using representative subareas over which the parameter values areeither constants, or the effects of subgrid heterogeneity are parameterized [5]. Themathematical approach described in this work along with a probabilistic model-ing of the inclusion’s spatial distribution or its electromagnetic parameters would
248
allows us to calculate the extreme (maximum or minimum) values of the averagecharacteristics of some anisotropic dielectric structures and, consequently those ofthe scattered fields. The efficiency of the method is demonstrated on an exampleof studying the average dielectric permittivity of a quasi-regular bulk structureformed by two sub-gratings with different geometric parameters and two differenttype of particles placed in its scattering nodes. One can analyze a simple three-phase structure which represents a mixture of dielectric particles (of the first type)and perfectly conducting particles (of the second type) arranged in a nonmagneticmedium to model some soil mixtures. Our recent results of the statistical modelingof dense anisotropic dielectrics whose scattering centers are distorted particles ofrevolutions may be used to study some ground covers and old ice. There is a stablegrowing interest in using microwave remote sensing techniques to monitor groundcovers, and, in particular, the changing sea ice covers in the polar regions. Someradars produce high-resolution images on a near-global scale providing a promis-ing technique for measuring many ice parameters of interest. There is, however,still a lack of understanding of how variations in the backscattering coefficientsare related to temporal and spatial variations in the physical parameters of seaice covers. It seems to be promising to incorporate this model into the finite mix-ture density model, and apply it to the classification of radar images from certainterrestrial covers.
References1. Khizhniak N. A. Integral Equations of Microscopic Electrodynamics,
Kiev, Naukova Dumka, 1986 (in Russian).2. Skirta E. A., N. A. Khizhniak, 1978. Some Problems of Theory of Gy-
rotropic Dielectrics, Izvestiya VUZov, Radiophysics, 22, 12, p. 1426.3. Avissar, R., 1992. Conceptual Aspects of a Statistical-Dynamical Ap-
proach to Represent Landscape Subgrid-Scale Heterogeneities in AtmosphericModels. JGR 97 (D3), p. 2729.
4. Boulet, G., J.D. Kalma, I. Braud, and M. Vauclin, 1999. An Assessmentof Effective Land Surface Parameterization in Regional-Scale Water Balance Stud-ies, J. Hydrology, 217, 225.
5. Albert, M.R., 2000. Notes on Current Techniques in Modeling SpatialHeterogeneity. 5th Eastern Snow Conference, Syracuse, NY, USA.
Department of Mathematics, East Stroudsburg University,East Stroudsburg, PA 18301, USAe-mail: [email protected]
249
Are stylized facts irrelevant in option pricing?
Tommi Sottinen, Finland
The classical Black-Scholes pricing model implies that the log-returns ofthe stock are independent and Gaussian. However, there is much empirical evi-dence that the log-returns are neither independent nor Gaussian. Moreover, manyeconomical time series show such stylized facts as long-range dependence, self-similarity, and power law of the log-returns. These stylized facts are not easily (ornot at all) captured by semimartingale pricing models such as the Black-Scholesmodel. So, there is need for non-semimartingale pricing models.
Non-semimartingale pricing models are not popular in stochastic finance. Thereason is the Fundamental Theorem of Asset Pricing: The pricing model admits ar-bitrage opportunities if and only if the stock-price process is not a semimartingale.So, as the no-arbitrage condition is an economic axiom, the non-semimartingalemodels are ruled out as pricing models by the fundamental theorem. However,by restricting the class of so-called admissible trading strategies one can excludethe arbitrage opportunities in many non-semimartingale models. Moreover, therestriction can be done in such a way that the replications for relevant optionsremain in the restricted class. For details on this we refer to [1].
In this talk we show how to incorporate some stylized facts (in particularthe long-range dependence and power laws) into the classical Black-Scholes pric-ing model. The resulting model will, in general, be a non-semimartingale model.However, if we restrict the class of admissible trading strategies there will be noarbitrage opportunities. Moreover, we shall see that the replication prices of theoptions turn out to be independent of the incorporated stylized facts. Actually,the only thing that determines the option prices will be the so-called pathwisequadratic variation. Now, the pathwise quadratic variation is a path propertythat that does not tell much (if anything) about the probabilistic structure of thestock-price process. In particular, the pathwise quadratic variation does not needto have anything to do with the variance of the log-returns.
We conclude that the Black-Scholes pricing model is surprisingly robust. Theprices it predicts can very well be correct even if the model is not. Indeed, as faras option pricing is concerned the stylized facts are mostly irrelevant.
[1] Bender, C., Sottinen, T., and Valkeila, E. (2006) No-arbitrage pricing be-yond semimartingales. WIAS Preprint No. 1110. Available electronically fromhttp://mathstat.helsinki.fi/∼tsottine/research.html .
University of Helsinki,Department of Mathematics and Statistics,PO Box 68, 00014 University of Helsinki, Finlande-mail: [email protected]
250
Monitoring Changes in Linear Models
Josef G. Steinebach, Germany
We consider the following linear regression model:
yi = xTi βi + εi (i = 1, 2, . . .),
with xi a p× 1 r.v., βi a p× 1 parameter vector, and εi a sequence of i.i.d.errors with mean 0 and variance σ2, satisfying certain additional assumptions.
On assuming that there is “no change” in a historical data set of size m, i.e.that βi = β0 (i = 1, . . . ,m), we are interested in testing (sequentially) thehypotheses
H0 : βi = β0 ∀ i vs. HA : βi = β0 ∀ i < m+ k∗, βi = β∗ ∀ i ≥ m+ k∗,
where β0 6= β∗ and k∗ (≥ 1) are unknown parameters.In Horvath et al. (2004), we propose two classes of monitoring schemes to
detect a structural change in the above model after the training period of sizem. The schemes are based on detector statistics Q(m, k) and boundary functionsg(m, k). We suggest to stop and reject H0 at
τ(m) = inf k ≥ 1 : Q(m, k) ≥ g(m, k) (inf ∅ = ∞).
Following Chu et al. (1996), our detectors are based on CUSUM’s of residuals, i.e.on bεi = yi −xTi bβm (i = m+ 1,m+ 2, . . .), with bβm denoting the least squaresestimator, based on y1, . . . , ym , of the in-control parameter β0. More precisely,
Q(m, k) = bQ(m, k) =
m+kXi=m+1
bεi , g(m, k) = c√m“
1 +k
m
”“ k
m+ k
”γ,
with c = c(α) (fixed). The weight functions g = g(m, k) can be chosen in aflexible way, depending on the parameter 0 ≤ γ < 1
2, according to whether an
“early” or “late” change after time m is expected. The procedures are designedsuch that the tests have an asymptotic (small) probability α of false alarm (asm→∞), i.e.
limm→∞
P˘τ(m) <∞
¯= α , under H0
(and asymptotic power 1, under HA). We have the following null asymptotic:
Theorem 1 (Horvath et al. (2004)) Under H0 ,
limm→∞
Pn 1bσm sup
k≥1
| bQ(m, k)|√m“
1 + km
”“k
m+k
”γ ≥ co
= Pn
sup0<t≤1
|W (t)|tγ
≥ co,
251
with W (t) : t ≥ 0 denoting a (standard) Wiener process and bσ2m a suitable
estimate of the unknown variance σ2.
A similar result can be obtained for a monitoring scheme based on CUSUM’sof recursive residuals eεi = yi − xTi bβi (i = m+ 1,m+ 2, . . .).
According to Horvath et al. (2004), if the change occurs shortly after themonitoring has commenced, then γ should be chosen as close to 1
2as possible
in order to minimize the detection time. Note, however, that γ = 12
is excludedfrom Theorem 1. We show that an extension to the boundary case is possible, ifwe choose a truncated modification of the stopping time τ(m) above, i.e.
τ∗(m) = inf k : 1 ≤ k ≤ N , bQ(m, k) ≥ c(m, t) g∗(m, k)
where inf ∅ = N = N(m), and
g∗(m, k) =√m“
1 +k
m
”“ k
m+ k
” 12, c(m, t) =
t+D(logm)
A(logm),
with A(x) =√
2 log x , D(x) = 2 log x + 12
log log x − 12
log π . Observe that, forγ = 1
2, the critical value c(m, t) increases to +∞ now like
√2 log logm. We
obtain:
Theorem 2 (Horvath et al. (2005)) Under H0, for all −∞ < t <∞,
limm→∞
Pn 1bσm max
1≤k≤N
| bQ(m, k)|g∗(m, k)
≥ c(m, t)o
= 1− exp˘− e−t
¯.
Under HA , in addition to the null asymptotics of Theorems 1 and 2, someresults on the asymptotic normality of the stopping times τ(m) and τ∗(m) canalso be established (cf., e.g., Aue et al. (2006)).
References
Aue, A., Horvath, L., Kokoszka, P., Steinebach, J. (2006) Monitoring shiftsin mean: Asymptotic normality of stopping times. Preprint, University ofUtah, Utah State University, and University of Cologne.
Chu, C.-S.J., Stinchcombe, M., White, H. (1996) Monitoring structural change.Econometrica, 64, 1045–1065.
Horvath, L., Huskova, M., Kokoszka, P., Steinebach, J. (2004) Monitoringchanges in linear models. J. Statist. Plann. Infer., 126, 225–251.
Horvath, L., Kokoszka, P., Steinebach, J. (2005) On sequential detection of para-meter changes in linear regression. Preprint, University of Utah, Utah StateUniversity, and University of Cologne.
Universitat zu Koln, Mathematisches Institut,Weyertal 86–90, D–50931 Koln, Germanye-mail: [email protected]
252
Invariance Principle for Diffusions in RandomEnvironment
Sven Struckmeier, Germany
We have shown an invariance principle for a diffusion process describing thediffusive motion of a particle x interacting with other particles randomly spreadedover the Euclidean space Rd .
The particles on Rd can be described by a so called configuration γ on R ,i.e., γ is a locally finite subset on Rd . Every configuration γ can be identifiedwith a positive Radon measure via γ ≡
Py∈γ δy, where δy is the Dirac measure
with mass in y . In this way, the set Γ = Γ(Rd) of all configurations is naturallytopologized with the weak topology. In particular, we have a Borel σ -algebraB(Γ) on Γ . Then, the configuration representing the particles will be chosen withrespect to a grand canonical Gibbs measure on (Γ,B(Γ)) . For details on Gibbsmeasures on Γ see, e.g., [1].
The particle x interacts with the particles y from the configuration γ via asymmetric translation-invariant pair-potential V : Rd → R with V (x) → +∞ forx → 0 (”repulsion“) and V (x) → 0 for |x| → +∞ (”decay at infinity“). Then itsmotion should satisfy the following SDE(
dXt = βγ(Xt) dt+√
2 dWt,
X0 = x0 (∈ Rd \ γ),(1)
where βγ(x) := −Py∈γ ∇V (x − y) and Wt is a standard Brownian motion.
Such diffusion equations with a singular drift have been solved for a large class ofpotentials V even in the strong sense in [3].
We have used an approach from [2] to show an invariance principle for solu-tions of the SDE (1), i.e., the time-space-scaled process Xε
t := εXε−2t convergesto a Brownian motion for ε → 0 . In this approach one considers the environ-ment process ξγt := y − Xt : y ∈ γ instead of the original process, i.e., themovement of the environment around the fixed point x . Averaging over the initialconfiguration γ with respect to a grand canonical Gibbs measure correspond-ing to the potential V , which is invariant and ergodic with respect to shiftsγ 7→ y + x : y ∈ γ , we could prove the conditions from [2] to obtain theinvariance principle for Xt .[1] S. Albeverio, Yu. G. Kondratiev, M. Rockner, Analysis and Geometry on Configu-
ration Spaces: The Gibbsian case, J. Funct. Anal. 157 (1998) 242-291.[2] A. De Masi, P. A. Ferrari, S. Goldstein, W. D. Wick, An Invariance Principle for
Reversible Markov Processes. Applications to Random Motions in Random Envi-ronments, J. Statist. Phys. 55 (1989) 787-855.
[3] N. V. Krylov, M. Rockner, Strong solutions of stochastic equations with singulartime dependent drift, Probab. Theory Relat. Fields 131 (2005) 154-196.
University of Bielefeld, GermanyUniversitat Bielefeld, Forschungszentrum BiBoSz. Hd. Sven Struckmeier, Postfach 10 01 31, D-33501 Bielefeld, Germanye-mail: [email protected]
253
On central moments of counting process
Olena Sugakova, Ukraine
Let ξi, i ≥ 1 be a sequence of random variables (r.v.) with equal expecta-tions Eξi = m > 0 . Assume that Eξpi <∞ for all i ≥ 1 for some p ≥ 2 . In thispresentation we don’t assume, that ξi are nonnegative.
Consider a counting process, generated by sequence ξi :
ν(t) = minn :
nXi=1
ξi > t;
and exceed over the boundary (residual life time) process:
ζt =
ν(t)Xi=1
ξi − t.
Let H(t) = Eν(t) be the renewal function and D(t) = Var ν(t) be thevariance of the counting process.
Lemma 1 Let ξi be a sequence of independent identically distributed r.v. Denote
ξ+ =
ξ1, if ξ1 ≥ 0;0, if ξ1 < 0.
Then
E (ν(t)−H(t))p ≤ 2p−1
mp
„2pp+ 2
p+ 1
E(ξ+)p+1
m+
+c12p−1(E|ξ|p +mp)H(t) + c2(Eξ2 +m2)Eν(t)p2
”,
wherec1 = 2 · 4p;
c2 = 24p2−p−2
p pp−3(p− 2)p−2
p
For p = 2 :
D(t) ≤ 2
m2
„16
3
E(ξ+)3
m+ 2(Eξ2 +m2)H(t)
«≤ 4
m3
„(Eξ2 +m2)t+
8
3E(ξ+)3+
+(E(ξ)2 +m2)E(ξ+)2´.
Lemma 2 Assume that ξi are independent but not necessary identically dis-tributed r.v. with E = m > 0 . Then the following relation holds:
E (ν(t)−H(t))p ≤ 2p−1
mp
`2pH(t)e+p + c12p−1(e+p +mp)H(t)+
254
+c2(supnEξ2i +m2)p2 Eν(t)
p2
”,
when e+p = supn≥1 E(ξ+n )p , and c1 , c2 are defined in Lemma1.These results can be used for estimations of large deviations probabilities and
in the theory of summation of random number of r.v.Let ν > 0 be a r.v.independent from ξi with natural values and distribution
Pν = n = pn, n ≥ 0 , such that Eν2 <∞ . Let
τ =
νXi=1
ξi.
We are interested in the behaviour Pτ > x as x→∞ .
Theorem 1 Let p(x) be a convex differentiable nonincreasing function, such thatp(n) = pn;n ≥ 0 . Denote R(x) = −p′(x) . Then for any α ∈ (0, 1) the nextinequality holds:
|Pτ > x −Pν > H(x)| ≤ 1
2R(αH(x))(D(x) + 1)+
+(2 + (H(x) + 1)p(H(x)))Pν ≤ αH(x)+ p(H(x)) + p(αH(x)) ≤
≤ 2p(αH(x)) +3 + 2Eν(1− α)2
D(x)
H2(x)+
1
2R(αH(x))(D(x) + 1).
References
[1] J.T.Chang, Inequalities for the overshoot, The Annals of Applied Probability4 (1994), N 4, 1223–1233.
[2] G.Lorden, On excess over the boundary, The Annals of Math. Statistics 41(1970), N 2, 520–527.
[3] O.V. Sugakova, Central moments of a nonhomogeneous renewal process,Theor. Probability and Math. Statist. 62 (2001), 157–163.
[4] A.A. Borovkov, S.A.Utev, Estimates for the distributions of sums stopped ata Markov moment , Teor. Veroyatnost. i Primen. 38 (1993), N 2, 259–272;English transl. in Theory Probab. Appl. 38 (1994).
Kyiv National University,Department of Mathematics, Faculty of Radiophysics,Volodymyrska 64, 01033, Kyiv, Ukrainee-mail: [email protected]
255
Non-Gaussian Long Range DependenceGyorgy Terdik, Hungary
It is a well known fact that for an i.i.d. series X1, X2, ...Xn (with meanzero and variance Var (Xk) = σ2
X ) the variance of the sum of Xk ’s changeslinearly with n . Asymptotically this result holds even when Xt is a stationarytime series. Now, if one estimates the variance of sums of a time series for differentn (usually called as aggregated series) then it is expected that the logarithm ofthese variances plotted against log (n) is linear with slope 1 . There are severalphenomena where it has been found that the slope is different from one, usuallyit is greater than 1 , i.e. for large n
log Var`Pn
k=1Xk´' 2H log (n) + const.,
where H ∈ (1/2, 1) . The half-slope H is called the Hurst coefficient, the namegoes back to Hurst (1951), we shall use a shifted version h = H−1/2 of H , sinceit is frequently used in time series analysis. This implies that the classical centrallimit theorems do not hold any more for these types of processes [4]. A stationarytime series Xt , t = 0,±1,±2...± n will be called long range dependent (LRD) ifits spectrum S2 (ω) behaves like |ω|−2h at zero, more precisely
limω→0
S2 (ω)
|ω|−2h= const., h ∈ (0, 1/2) . (1)
This definition of long range dependence can be stated in terms of autocorrelationfunction, because (1) is equivalent to:
limk→∞
Cov (Xt, Xt+k)
|k|2h−1= const., h ∈ (0, 1/2) . (2)
In other words, the autocorrelation function decays hyperbolically. In fact al-though the spectrum is in L1 its Fourier coefficients Cov (X0, Xk) are not inL1 any more. The equivalence of (1) and (2) based on theory of regular vary-ing functions, [1]. Let L (x) be quasi monotone slowly varying function, then forh ∈ (0, 1/2)
∞Xk=0
L (k) |k|2h−1 ei2πωk ' 2 |2πω|−2h L (1/ |ω|) Γ (2h) cos (πh) , ω → 0,
i.e. the left hand side converges to a function, S2 (ω) say, such that it fulfils (1),[6]. The Tauberian conversion of this Abelian result is also true. These results ofcalculus covers totally the problem of LRD for Gaussian time series. If the timeseries is not Gaussian then the higher order cumulants and spectra provide someadditional information on the structure of dependence.
Let the process Xt be stationary in third order and centered, denote
Cum (Xt+k1 , Xt+k2 , Xt) = c3 (k1, k2) , k1, k2 = 0, 1, 2, . . . .
256
its third order cumulants. The bispectrum S3 is a complex valued function of twovariables with Fourier coefficients c3(k1, k2) , i.e.
c3 (k1, k2) =
1/2ZZ−1/2
ei2π(ω1k1+ω2k2)S3 (ω1:2) dω1:2.
The relations (1) and (2) concern on the behavior of the covariances and thespectrum. The bispectrum S3 (ω1:2) and the third order cumulant are connectedin similar way. Let αω2/ω1 = arctan (ω2/ω1) , α ∈ (0, π/4) .
DefinitionThe time series Xt is long range dependent in third order with radialexponent χ and spherical exponents χ1, χ2 if the bispectrum S3 is factorized as
S3 (ω1:2) ' const. |ω1:2|−3χ α−χ1ω2/ω1
˛π/4− αω2/ω1
˛−χ2 Lα`|ω1:2|−1´L(s)
“α−1ω2/ω1
”,
(3)and 0 < χ < 2/3 , χ1, χ2 ∈ [0, 1) . L(s) is some spherical slowly varying functionwith the form L(s)
“α−1ω2/ω1
”= L
“α−1ω2/ω1
”L“˛π/4− αω2/ω1
˛−1”, and Lα is a
slowly varying family.
If the spherical part is bounded, i.e. χ1 = χ2 = 0 , then the series of thirdorder cumulants c3 (k1, k2) = Cum (Xt+k1 , Xt+k2 , Xt) diverges with order 3χ−2 ,
more precisely ifpk22 + k2
1 is large then |c3 (k2, k1)| '“p
k22 + k2
1
”3χ−2
, see [5],Ch. VII. Theorem 2.17. The only stochastic process having such type "isotropic"bispectrum, we have our mind, is considered in [3].
In this paper we show particular examples of time series with third order LRD,and highlight some connections to the asymptotic self-similarity in third order.References
1. N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular variation, Encyclopediaof Mathematics and its Applications, vol. 27, Cambridge University Press, Cam-bridge, 1987.
2. H.E. Hurst, Long term storage capacity of reservoirs, Trans. Am. Soc. Civil Engrs.,116 (1951), 770-808.
3. E. Igloi and Gy. Terdik, Superposition of diffusions with linear generator and itsmultifractal limit process, ESAIM Probab. Stat. 7 (2003), 23-88(electronic).
4. P. Major, Multiple Wiener-Ito integrals, Lecture Notes in Mathematics, vol. 849,Springer-Verlag, New York, 1981.
5. E.M Stein and G. Weiss, Introduction to Fourier analysis on Euclidian spaces,Princeton University Press, Princeton, N.J., 1971, Princeton Mathematical Series,No. 32.
6. M.S. Taqqu, Fractional Brownian motion and long-range dependence, Theory andapplications of long-range dependence, Birkhauser Boston, Boston, MA, 2003, pp.5-38.
University of Debrecen,H-4010 Debrecen Pf. 12, Hungary
257
Small deviations for Non-Gaussian processes
Simon Thomas, France
Small deviations consists in studying the asymptotics of logP (‖X‖ < e) whene tends vers 0, for given process X and norm ‖.‖ . This kind of results is inter-esting in its own right, but may also be useful for other questions in analysis,probability or statistics. Contrary to large deviations, the speed of convergencedepends often on several parameters, such as the regularity of the process. Thereexists already a rich literature in this field for Brownian motion or more gen-eral Gaussian processes. I will present some results in a Non-Gaussian framework,paying a special attention to stable processes.
Departement de Mathematiques,Universite d’Evry-Val d’Essonne,F-91025 EVRY Cedex, FRANCEe-mail : [email protected]
258
On pasting two diffusion processes on asmooth hypersurface with a general adjunction
Wentzel conditionZhanneta Tsapovs’ka, Ukraine
Let Rn , n > n , be a real n -dimensional Euclidean space in which a smoothclosed hypersurface S is given that subdivides Rn into two domains: the internalD1 and the external D2 = Rn \ D2 , where D1 = D1 ∪ S .
We assume that in the domains D1 and D2 noninterrupted diffusion processesare given that are ruled by the second order generated differential elliptic opera-tors L1 and L2 respectively whose coefficients are bounded continuous functionson Rn and their diffusion matrices are symmetric, positively determined and uni-formly nondegenerated. Suppose also that at the points of S a Wentzel typeadjunction operator L0 is given [1] which is responsible for all the possible exten-sions of the diffusion processes in the case of their outcome on S . We will assumethat the coefficients of L0 are bounded continuous functions on S , the operatorL0 is uniformly elliptic in tangent variables and, moreover, all the points of theboundary S has the properties of delay and partial reflection.
A problem is posed on existence of the semigroup operators Tt , t > 0 , thatdescribes a noninterrupting continuous Feller process in Rn such that its partsin Di , i = 1, 2 , coincide with the diffusion processes and its behavior on S isdetermined by the given adjunction operator L0Z . The given problem is alsocalled the problem of pasting two diffusion processes and also the problem onconstruction of a mathematical model of the diffusion phenomenon in Rn wherea membrane is located on S (see [2]).
We construct the required semigroup by means of the methods of classicalpotential theory as a solution of the corresponding parabolic adjunction problem.Here, we additionally assume that the coefficients of the operators Li , i = 1, 2 ,and L0 are Holder with some index λ , λ ∈ (0, 1) , respectively on Rn and S andthe surface S belong to the Holder class H2+λ . It is also proved that, for theobtained process, there exist diffusion coefficients, in the usual sense, which arepiesewise continuous functions defined on Rn .
References
[1] A. D. Wentzel, On boundary conditions for multidimensional diffusionprocesses. – Teor. veroyatn. i ee prim. 1982. – v. 4, N 2. – P. 172–185.
[2] M. I. Portenko, Diffusion processes in media with membranes. – Proc. of In-stitute of Mathem. of National Academy of Sci. of Ukraine, Kyiv, 1995.
Department of Mathematics,Ivan Franko Lviv National University,Universytetska 1, Lviv, 79000, Ukrainee-mail: [email protected]
259
A problem of distinguishing between pointlight sources
Valery N. Turchyn, Ukraine
A problem of distinguishing and detection of objects using optoelectronic sys-tem (OES), in particular, a problem of distinguishing between two light sources,is important among the problems of Earth remote sensing by a spacecraft (SC).
A photoreceiving device of OES is a device with charge link (DCL-ruler).DCL-ruler is a segment consisting of pixels on which charges accumulate.
An image of the Earth area being shot is formed by consequent reading ofinformation from DCL-ruler (bar) which moves perpendicularly to lane of SCduring its movement along the orbit. Information from DCL-ruler is read afterequal time intervals, length of each of these intervals is equal to exposure time.
Two light sources are distinguishable if they get on different pixels duringmovement of DCL-ruler, otherwise two light sources are classified as one lightsource. But during movement of DCL-ruler radiation from light source gets ontwo pixels (an image "blur" takes place). Therefore two light sources can be dis-tinguished if they lighten at least three pixels along the direction of SC’s movementor at least four pixels along the direction different from SC’s movement direction.
It’s necessary to take into account while solving the problem of distinguishingbetween light sources that the following values are random: 1) distance ξ from thenearest edge of left (with respect to light source) pixel’s trace on Earth to lightsource, 2) angle ϕ between the SC’s movement direction and direction of segmentwhich connects light sources, 3) distance η between light sources. Therefore whenwe talk about distinguishing between light sources it’s necessary to talk aboutprobability of distinguishing between them.
If distributions of random variables ξ , ϕ , η are given and if it’s supposed thatduring the exposure time trace moves along the Earth surface in such way thatits left (with respect to direction of movement) edge coincides with the right edgein the previous exposure, probabilities of distinguishing between two light sourcescan be calculated.
ReferencesE.D. Yarmolchuk. Informational characteristic of video section of Sich-2 spaceEarth observation system with spacecraft MS–2–8// Collected scientific works ofZhVIZE. – 2004. – Vol. 8. – Zhitomir.
Dnipropetrovsk National University,faculty of mechanics and mathematics,department of statistics and probability theory,Naukova str., 13 Dnipropetrovsk, 49050, Ukraine
260
Glauber and Kawasaki dynamics forDeterminantal Point Processes
Nataliya Turchyna, Germany
We construct Glauber and Kawasaki equilibrium dynamics of infinite parti-cle systems in a Riemannian manifold X , for which certain determinantal pointprocesses are invariant.
Let X be a connected oriented C∞ manifold. Let B(X) denote the Borelσ -algebra on X and m the volume measure on X .
The configuration space Γ := ΓX over X is defined as the set of all subsetsof X which are locally finite:
Γ :=˘γ ⊂ X : |γ ∩ Λ| <∞ for each compact Λ ⊂ X
¯,
where | · | denotes the cardinality of a set. One can identify any γ ∈ Γ withthe positive Radon measure
Px∈γ εx ∈ M(X) , where εx is the Dirac measure
with mass 1 at x ,Px∈∅ εx:=zero measure, and M(X) stands for the set of all
positive Radon measures on B(X) . The space Γ can be endowed with the relativetopology as a subset of the space M(X) with the vague topology. We shall denotethe Borel σ -algebra on Γ by B(Γ) .
A measure µ on (Γ,B(Γ)) is said to have correlation functions if for anyn ∈ N there exists a non-negative measurable symmetric function k
(n)µ on Xn
such that, for any measurable symmetric function f (n) : Xn → [0,∞] ,ZΓ
Xx1,...,xn⊂γ
f (n)(x1, . . . , xn)µ(dγ)
=1
n!
ZXn
f (n)(x1, . . . , xn)k(n)µ (x1, . . . , xn)m(dx1) . . .m(dxn).
Let K be a bounded Hermitian locally trace class integral operator on L2(X, dm) ,denote by K(x, y) the integral kernel of the operator K .
Definition A probability measure µ on (Γ,B(Γ)) with correlation function
k(n)µ (x1, . . . , xn) = det(K(xi, xj))
ni,j=1.
is called a determinantal point process with kernel K.
Theorem 1 ([4]) A Hermitian locally trace class integral operator on L2(X, dm)defines a determinantal point process µ if and only if
0 ≤ K ≤ I.
If the measure µ exists, then it is unique.
261
The Glauber dynamics is a special birth-and-death process in X , while in theKawasaki dynamics interacting particles randomly jump over X . The generator ofthe Glauber dynamics for continuous particle system in X is given by the formula
(HGF )(γ) =Xx∈γ
d(x, γ)(D−x F )(γ) +
ZX
b(x, γ)(D+x F )(γ) dx,
where
(D−x F )(γ) = F (γ \ x)− F (γ), (D+
x F )(γ) = F (γ ∪ x)− F (γ).
The coefficient d(x, γ) describes the rate at which the particle x of the configu-ration γ dies, while b(x, γ) describes the rate at which, given the configurationγ , a new particle is born at x .
The generator of Kawasaki dynamics for continuous particle system is givenby
(HKF )(γ) =Xx∈γ
ZX
c(x, y, γ)(F (γ \ x ∪ y)− F (γ)) dy,
and the coefficient c(x, y, γ) describes the rate at which the particle x of theconfiguration γ jumps to y .
Using the theory of Dirichlet forms [3], we establish conditions on symmetriz-ing measures and generators of both dynamics under which corresponding con-servative Markov processes exist. Our result is based on the ideas of the articles[1, 2], where Glauber and Kawasaki dynamics of continuous particle systems ininfinite volume, which have a Gibbs measure as a symmetrizing measure, are con-structed. We prove also the coercivity identity for the generator of the Glauberdynamics, find a sufficient condition for the existence of the spectral gap, and givesome examples when these conditions are satisfied.
References
[1] Yu. G. Kondratiev, E. Lytvynov, Glauber dynamics of continuous particlesystems, Ann. Inst. H. Poincarй Probab. Statist. 41 (2005), no. 4, 685–702.
[2] Yu. G. Kondratiev, E. Lytvynov, M. Rockner, Equilibrium Glauber andKawasaki dynamics of continuous particles systems, preprint.
[3] Z.-M. Ma, M. Rockner, An Introduction to the Theory of (Non-Symmetric)Dirichlet Forms, Springer-Verlag, 1992.
[4] A. Soshnikov, Determinantal Random Point Fields, Russ. Math. Surv. 55(2000), 923-975.
Fakultat fur Mathematik, Universitat Bielefeld,Postfach 10 01 31, D-33501 Bielefeld, Germanye-mail: [email protected]
262
On the ruin probability of the insurancecompany when the claims sizes distributed as
a mix of shifted exponential distributionsArtem Tymko, Ukraine
Consider a Sparre Andersen risk process R(t) = u + ct −PN(t)j=1 Yi defined
in terms of the following values: u = R(0) > 0 - is the initial risk reserve; notdecreasing sequence of random variables t0 = 0 ≤ t1 ≤ t2 ≤ . . . - the mo-ments of approach of each claim; variables τn = tn − tn−1 are the interclaimtime; Fτi(x) = Fτ (x) denotes function of distribution of random variables τi ;N(t) = supn : tn ≤ t denotes number of claims having occurred up totime t ≥ 0 ; sequence of the independent identically distributed random vari-ables Ynn≥1 is the sequence of the claims sizes at the moment tn ; c > 0 is thepremium received continuously per unit time.
Suppose that Ynn≥1 and τnn≥1 are independent, and the relative securityloading c Eτ1
EY1− 1 > 0 . Let the claim sizes Ynn≥1 have density of distribution
f(y) =PNk=1 pkfk(y) , where 0 ≤ pk ≤ 1 ,
PNk=1 pk = 1 and fk(y) =
=
λke
−λk(y−ak), y ≥ ak0 , y < ak
, 0 < λ1 < λ2 < · · · < λN , 0 < a1 < a2 < · · · < aN .
Denote by ψ(t, u) the probability of ruin within finite time and byγ(α) =
R∞0e−αtFτ (dt) the Laplace transform of Fτ (t) .
In the given denotations and suppositions the following theorem is true.
Theorem 1 The Laplace transform for the ruin probability of the insurance com-pany by the moment of time t is equal:Z ∞
0
e−αtψ(t, u)dt =(γ(α)− 1)(1−
PNk=1 pke
λkakgk(α))
α(1− γ(α)PNk=1 pke
λkak ),
where gk(α) are the unique solutions of the system of linear equations:0BB@a11 a12 . . . a1N
a21 a22 . . . a2N
. . . . . . . . . . . .aN1 aN2 . . . aNN
1CCA0BB@g1(α)g2(α). . .
gN (α)
1CCA =
0BB@e1e2. . .eN
1CCA .
Here ei = e−βi(α)uQNk=1(λk − βi(α)) , aik = pkλke
λkakQNj=1,j 6=k(λj − βi(α)) ,
where βi(α) : 0 < β1(α) < λ1, λ1 < β2(α) < λ2, . . . λN−1 < βN (α) < λN arezeroes of
Q(β) =
NYk=1
(λk − β)− γ(α+ cβ)
NXk=1
0@pkλkeλkak
NYj=1,j 6=k
(λj − β)
1A .
Donetsk National University,Universitetskaya str., 24, 83055 Donetsk, Ukrainee-mail: [email protected]
263
Random symbolic dynamic systems andadditive multi-step Markov chains
Oleg Usatenko2 , Valery Yampol’skii2 , Sergej Melnyk 2 ,Stanislav Apostolov1 , Zakhar Mayzelis1 , and Konstantin Kechedzhy2 , Ukraine
The problem of systems with long-range spatial and/or temporal correlations(LRCS) is one of the topics of intensive research in modern physics, as well as inthe theory of dynamic systems and the theory of probability.
In Ref. [1] the concept of additive Markov chain with the step-wise memoryfunction was introduced. This allowed us to find analytically the probabilities ofthe words of length less than N occurring. A self-similarity of the studied sto-chastic process is revealed and the similarity group transformation of the chainparameters is presented. The diffusion Fokker-Planck equation governing the dis-tribution function of the L -words is explored. If the persistent correlations arenot extremely strong, the distribution function is shown to be the Gaussian withthe variance being nonlinearly dependent on L . We prove that this distributionobeys the power law with the exponent of the order of unity in the case of ratherstrong persistent correlations. The applicability of the developed theory to thecoarse-grained written and DNA texts is discussed.
There exist some dynamic systems (coarse-grained sequences of DNA andliterature texts, and non-extensive Ising chains of spins) having the correlationproperties that can be adequately described by the additive Markov chains withthe memory function that differs from step-wise one. We determine the binaryN -step Markov chain as a sequence of symbols ai = 0, 1 , with the conditionalprobability function of additive form,
P (ai = 1 | ai−N , ai−N+1, . . . , ai−1) = a+NPr=1
F (r)“ai−r − a
”,
where F (r) is the memory function and a is the average relative number of unitiesin the chain.
The algorithm for finding the correlation functions Ks(r1, . . . , rs−1) of sthorder is proposed. An equation connecting the memory and correlation functionsis obtained:
Ks(r1, . . . , rs−1) =NPr′=1
F (r′)Ks(r1, . . . , rs−1 − r′).
Together with the proper initial conditions this expression permits one toderive all the correlation functions sequentially. It also allows us to constructdirectly a sequence with prescribed correlation function K2(r) .
References
[1] O. V. Usatenko and V. A. Yampol’skii, Phys. Rev. Lett. 90, 110601 (2003).1 Kharkov National University, Svoboda Sq. 4, Kharkov 610772 A. Ya. Usikov Institute for Radiophysics and ElectronicsUkrainian Academy of Science, Proskura Str. 12, Kharkov 61085e-mail: [email protected]
264
Optimality of Corrected Score in a polinomialmeasurement error model
Olena Usoltseva, Ukraine
We consider a class of linear estimates of a parameter β in a polinomialregression model:
y =
mXi=0
βixi + ε,
where ε ∼ N(0, σ2u) with unknown variance σ2
u. Instead of x we observe w = x+u,where
x ∼ N(µx, δ2x), u ∼ N(0, σ2
u), and σ2u is known. Here ε, x and u are inde-
pendent.A linear score function has a form
SL(w, y, β) = P (w)y +Q(w)β,
where β = (β0, · · · , βm), ρ = (1, x, · · · , xm)T , E(P |X)ρT = E(Q|X), which im-plies E(SL|X) = 0. The estimator β is defined as a measurable solution to theequation
Pni=0 SL(wi, yi, β) = 0. The asymptotic covariance matrix (ACM) of
the estimator equals
ΣL = σ2εΦ(p) = (EpρTw)−1(EvppT + Cov(mp−Qβ))(EpρTw)−T ,
where ρw = E(ρ(x)|W ), and m and v are conditional mean and variance of Ygiven W . We form βT = [β0β
t−0]. From one hand ΣL does not depend on β0 ;from
another hand, the following bound is achieved:
minβ
ΣL = ΣL|β−0=0 .
Hereafter matrices are compared with respect to the Lowener order. The fol-lowing bound takes place:
ΣL|β−0=0 ≥ (EρwρTw)−1, (1)
and equality in (1) is attained if and only if p = Kρw, where K is non-random non-singular. Let us consider a particular case of SL , Corrected Score(CS): SCS = ρ−xy + (ρρT )−xβ, where ρ−x = ρ−x(w) is the only solution to thedeconvolution problem E(ρ−x|X) = ρ(x) in the class of tempered distributionsS′(R → Rn). Now, for the polinomial model ρ−x = Kρw,K is a non-singularmatrix. Then equality in (1) takes place for SL = SCS . Thus
ΣL|β−0=0 ≥ minβ
ΣCS . (2)
265
In this sense show the most efficient linear Score function is SCS . More-over if for certain linear Score function SL, equality in (2) is attained, thenSL and SCS generate the same estimator β = βCS . Besides this, we con-sider a general linear-in-β measurement error model y = ρTβ + ε, withρT (x) = (1, f1(x), · · · , fm(x)), where 1, f1, · · · , fm are linearly independent. Sim-ilarly we introduce linear Score SL. The equality in (1) is not always attained forsuch p that E(p(w)|X) = ρ(x). This is the case, e. g., for the exponential modelwith ρT (x) = (eλ0x, eλ1x, · · · , eλmx), λ0 = 0, λi 6= λj , i 6= j. We mention that theQuasi Score (CS) estimator is more efficient that the CS estimator, see Shklyar etal. (2005), but the QS function does not belong the class of our linear Scores.
References : S. Shklyar, H. Schneeweiss, and A. Kukush, Quasi Score is moreefficient than Corrected Score in a polinomial measurement error model. Discus-sion Paper 445, SFB 386. LMU, Munich, 2005. To apper in Metrica.
Kiev National Taras Shevchenko Universitye-mail: [email protected]
Random walks on graphs with memory andtheir cryptographical properties
Vasyl Ustimenko, Ukraine
The term graphs with memory used for an infinite family of finite graphsΓi(K) with the vertices and which are tuples over the alphabet K and the choiceof neighbour described by disjoint union of several Cartesian powers of K . Thefamily of graphs with memory can be treated as special models of Turing machinewith the internal and external alphabet K .
The diameter of a k -regular graph (or graph with the average degree k) oforder v is at least logk−1(v) and it is known that the random k -regular graphhas diameter close to this lower bound. In the case of family of small world graphsthe diameter is O(logk−1(v)) . The girth of the graph is the smallest length of itis cycle.
THEOREM: For each pair (k ≥ 3, g ≥ 3) there is a regular small world graphof degree ≥ k and girth ≥ g ,
The proof based on explicit constructions of infinite families of graphsCD(n, q) defined over the finite field Fq [1]. W. Kantor conjectured that theyare small world graphs more then 10 years ago. The justification of the abovetheorem obtained via the prove of Kantor’s conjecture.
The natural parametrisation of walks ( computations in the correspondingTuring machine) for graphs with memory CD(k, q) can be defined. Such walkscan be used as encryption tools (see [2], [3]), In particular, for design of streamciphers (case of Markovian Process) and symmetrical ciphers with nontriangular
266
encryption, public key algorithm [4]. The above result justify the possibility totransfer the plaintext into arbitrarily chosen ciphertext.
The symmetrical algorithms turns out to be robust, they have flexible keys.The software based on such idea first was implemented first for the UniversityNetwork of 13 countries in the South Pacific, later in Australia, Emirates, Canada,Ukraine. Such graphs can be used as well in Error Correction [4].
The talk is devoted to generalisations and modifications [5] of such algorithms(change of Fq on Zn in particular leads to faster encryption, theoretical estimatesof their speed of algorithms, evaluation of the encryption resistance to differentattacks of opponent, sophistication level of the "seeming chaos".
The algorithms can be used for the generation of new pseudo random se-quences.
References
[1] F.Lazebnik V.Ustimenko and A.J.Woldar A new series of dense graphs of highgirth , Bulletin of the AMS 32 (1) (1995), 73-79.
[2] V. Ustimenko, Graphs with Special Arcs and Cryptography, Acta ApplicandaeMathematicae, 2002, vol. 74, N2, 117-153.
[3] V. Ustimenko, CRYPTIM: Graphs as tools for symmetric encryption, In Lec-ture Notes in Comput. Sci., 2227, Springer, New York, 2001.
[4] V. Ustimenko, Maximality of affine group and hidden graph cryptsystems,Journal of Algebra and Discrete Mathematics, October, 2004, v10, pp 51-65.
[5] V. A. Ustimenko, Linguistic Dynamical Systems, Graphs of Large Girth andCryptography. Zapiski Nauchnyh Seminarov POMI, vol. 326, “RepresentationTheory, Dynamical Systems , Combinatorial and Algorithmic Methods", 2005,214-235.
[6] P. S. Guinard and J.Lodge, Tanner Type Codes Arizing from Large GirthGraphs, Communications Research Centre, Canada , Reprint GUI94, 2006.
Institute of telecommunications and global informational space, NAS Ukraine,and University of Maria Curie Sklodovska, Lublin, Polande-mail: [email protected]
267
Fibonacci representation of real numbers
N. Vasylenko, Ukraine
There exist different ways to name and to write numbers i.e., numerations(from latin numerus - number, numeration - the account). The task of numerationis a representation of any natural number using small group of special characters(symbols). There are many ways to represent the same number. So, there aremany numerations. It is necessary to distinguish between notion of numerationsystem and notion of numeration. The numeration system is a notion of theoreticalarithmetics. It has appeared rather recently and became common for scientistsof all ethnoses and nations. The used different numeration systems in differenthistorical phases. The Fibonacci sequence has arisen as a solution of a knownproblem „About rabbits” by Leonardo Pisano. The Fibonacci sequence is a basisof Fibonacci numeration system.
Theorem 1 Any natural number a can be represented in the form
a = fn · un + fn−1 · un−1 + · · ·+ fk · uk + · · ·+ f2 · u2,
where fn 6= 1, fi ∈ 0, 1, ui is i-th Fibonacci number, i = 2,∞ [1, p.105],[2, p.36-38], [4, p.117-119].
Theorem 2 Any real number x ∈ [0;S] can be represented in the form:
x =
∞Xi=2
fiui
=f2u2
+f3u3
+ · · ·+ fkuk
+ · · · , (1)
where fi ∈ 0, 1, i ∈ 2, 3, . . . , k, . . ., S =∞Pi=2
1ui
= 2, 35988566 . . . (Andre-
Jeannin R., 1989).
The expression (1) is called the Fibonacci representation of the number x. Wedenote the expression (1) briefly by x = ∆f2...fk....
Theorem 3 For any number x ∈ [0;S] there exist at least one sequence fk(x),fk(x) ∈ 0, 1 such that x = ∆f2(x)...fk(x)....
References
[1] J.-P. Allouche, J. Shallit, Automatic sequences: theory, applications, general-izations, Cambridge University Press, Cambridge, 2003, 588 p.
[2] N. N. Vorobjov, The Fibonacci number, Nauka, Moscow, 1969, 112 p.[3] M. V. Pratsiovytyi, Fractal approach to investigations of singular probability
distributions, National Pedagogical Univ., Kyiv, 1998, 296 p.[4] M. Yadrenko, Discrete mathematics: manual, PTandMS, Kyiv, 2004, 245 p.
268
Mixing for Markov and Markov conditionaldynamics
In memory of Professor Mikhail Yosifovich Yadrenko
Alexander Veretennikov, UK & Russia
1 Mixing for Markov chains and Ito’s SDEs, joint withS. Klokov
Consider a Markov chain from the class of non-linear autoregression processes,
(1) Xn+1 = Xn + b(Xn) + ξn+1, n ≥ 0,
where random variables (ξn, n ≥ 1) are IID, in a d -dimensional Euclidean space.We assume the following basic assumptions, which are often easy to check:
(2) lim|x|→∞
fib(x),
x
|x|1−p
fl= −r < 0, 0 ≤ p ≤ 1;
Eξn = 0 , and E exp(c|ξ|) <∞ with some c > 0 , for simplicity b is bounded.The Theorem states that for (Xn) , there are mixing bounds, and bounds for
convergence to equilibrium, exponential for p = 0 , polynomial for p = 1 dependingon r that has to be large enough, and sub-exponential for 0 < p < 1 . All boundsare precise in an appropriate sense, as shown independently by both authors.
The method is based on two ideas: recurrence bounds and Doeblin–Doob’s type condition. Similar results hold true for Ito’s equations,dXt = b(Xt) dt + dWt, X0 = x . Here the role of Harnack’s inequalities is thatthese provide Doob–Doeblin type conditions for “the process on some ball.” Mostappropriate for this method is β -mixing coefficient. In the case 0 < p < 1 , sub-exponential bounds were obtained by M. Malyshkin (2000), and then improvedby the authors. The cases p = 0 (exponential) and p = 1 (polynomial) have beenstudied in [4, 5].
2 Mixing for Ito’s SDE approximations, joint with S. Klokov
Similar results have been established for approximations of an SDE (1),
(3) Xh(n+1)h = Xh
nh + b(Xhnh)h+
√hξn+1.
There are basically two main models: “strong approximations” with Gaussian noiseξ , and “weak approximations” with non-Gaussian noise. For the first one, thescheme (3) possesses exactly the same upper bounds for mixing as the limitingprocess that solves the corresponding SDE, [2]. For the second one, similar boundshave been proved by the authors for certain classes of processes using differentmethods: one based on Malliavin’s calculus, [3], and the other one, “by hand”, onlocal theorem techniques, and on ideas close to N. V. Krylov – M. V. Safonov’s.Certain results will be presented on the case of variable “diffusion coefficient”, i.e.with the last term σ(Xh
nh)√hξn+1 in (3), which remains the most challenging.
269
3 Mixing for filtered Markov chains, joint with M. Kleptsyna
Suppose we do not see the signal (Xn) from (1), but observe the values
(4) Yn = h(Xn) + ηn, n ≥ 1,
where (ηn) is an IID sequence satisfying certain conditions; in particular, Gaussiancase is included. We consider optimal filter measures,
µYn (dxn) := Pµ0(Xn ∈ dxn | FYn ).where µ0 is the distribution of X0 . Then the simplest problem of mixing maybe formulated in the following way: suppose there are two copies of our filter,with two different initial distributions, µ0 and ν0 , denote them by µY,µ0
n andνY,ν0n . Naturally, there is only one observation process Y available. The questionsis whether their difference converges to zero in the long run:
(5) ‖µY,µ0n − νY,ν0n ‖TV → 0, n→∞.
So far, general results have been established only for the uniformly ergodic case.We present [1] the Theorem which states that in the case of exponential (non-uniform) recurrence for the signal process (X) , there is a convergence in (4), too,with another exponential rate; if the signal is recurrent with a “better than anypolynomial rate”, then there is a “better than any polynomial” convergence in (5),too. Similar results hold true for polynomial recurrence, and for diffusions.
References
[1] Kleptsyna, M. L., Veretennikov, A. Yu. On discrete time er-godic filters with wrong initial data. URL: www.math.univ-lemans.fr/prepublications/05/filre.pdf
[2] Klokov, S. A.; Veretennikov, A. Yu. On mixing rate for Euler scheme for sto-chastic difference equations, Dokl. Akad. Nauk (Doklady Acad. Sci. of Russia),69(2), 2004, 273–274.
[3] Klokov, S. A.; Veretennikov, A. Yu. Mixing and convergence rates for a familyof Markov processes appriximating SDEs. Random Oper. and Stoch. Equ.,14(2), 2006, 103–126.
[4] Veretennikov, A. Yu. Estimates of the mixing rate for stochastic equations.(Russian) Teor. Veroyatnost. Primenen. 32 (1987), no. 2, 299–308; Englishtranslation: Theory Probab. Appl. 32 (1987), no. 2, 273–281.
[5] Veretennikov, A. Yu. On polynomial mixing and the rate of convergence forstochastic differential and difference equations. Teor. Veroyatn. i Primenen.44 (1999), 2, 312–327; Engl. transl. in Theory Probab. Appl. 44 (2000), 2,361–374.
School of Mathematics, University of Leeds, UK,& Institute of Information Transmission Problems, Moscow, RussiaMailing Address: School of Mathematics, University of Leeds, LS2 9JT, UKe-mail: [email protected]
270
The statistical simulation of isotropic 3-Drandom fields with the Bessel’s correlation
function.
Zoya A. Vyzhva, Ukraine
Let ξ(x) (x = (r, θ, ϕ) , r ∈ R , θ ∈ [0, π] , ϕ ∈ [0, 2π]) is homogeneous andisotropic, continuous in the mean square, random fields on the space R3 with zeroexpectation and dispersion:
Eξ2(x) =
Z ∞
0
dΦ(λ),
where Φ(λ) is spectral function of random field ξ(x).We may represent this random field by the "spectral decomposition" (see [1]):
ξ(r, θ, ϕ) =
∞Xm=0
mXl=0
Cm,lPlm (cos θ)
"cos lϕ
Z ∞
0
Jm+ 12(λr)
√λr
Zlm,1(dλ)+
+ sin lϕ
Z ∞
0
Jm+ 12(λr)
√λr
Zlm,2(dλ)
#, (1)
where
Cm,l =
sπ
2νl(2m+ 1)
(m− l)!
(m+ l)!, νl =
1, l = 0;2, l > 0.
P lm(t) is a joint Legendre functions , Jn(t) is the Besssel function of the first kindand
˘Zlm,p(.)
¯∞,mm=0,l=0
(p = 1, 2) are sequences of orthogonal random measures onσ -algebra of Borel subsets from the interval [0,+∞) , so, that:
EZlm,k(S1)Z lm,k(S2) = δmmδllδkkΦ(S1
\S2), (2)
∀S1, S2 ∈ B([0,+∞)).
We used estimator of this random field approximation by partial sums ofspecial kind. We considered the following division of the interval:
[0,+∞) =
k[i=1
Λi =
k−1[i=1
Λi[
Λk,
where the domains Λi = λ : ai ≤ λ < ai+1 (ai ∈ R, i = 1, k − 1) have the fi-nite diameter di(i = 1, k − 1) and domain Λk = λ : λ ≥ ak have the infinitediameter. We choose the point-variable λi in every domain Λi .
271
Than the partial sums of special kind are the statistical model of 3-D randomfield:
ξNk (r, θ, ϕ) =
kXi=1
NXm=0
mXl=0
Cm,lPlm (cos θ)
"cos lϕ
ZΛi
Jm+ 12(λir)
√λir
Zlm,1(dλ)+
+ sin lϕ
ZΛi
Jm+ 12(λir)
√λir
Zlm,2(dλ)
#, (3)
We used the algorithm of statistical simulation of the 3-D random fields , whatwas construct by means the mean-square estimator(see [2]):
EZ|x|≤Q
hξ(x)− ξNk (x)
i2dx
where Q is some real nonnegative number (radius of sphere).We used the randomization method .We constructed this algorithm for the generation of random isotropic 3-D field
ξ(x) realizations with the Bessel’s correlation function:
B(ρ) = 3
rπ
2
J 32(aρ)
(aρ)32, a > 0.
References
1. M. Yadrenko. Spectral Theory of Random Fields. Optimization SoftwareInc. New York (1983).
2. Z. Vyzhva. About Approximation of 3-D Random Fields and StatisticalSimulation. Random Operators and Stochastic Equations. Vol. 4, No.3, 255-266(2003).
Department of General Mathematics,Faculty of Mechanics and Mathematics,Kiev National Taras Shevchenko University,Volodymyrska 64, 01033, Kyiv, Ukrainee-mail: [email protected].
272
Stochastic processes in Sobolev spaces.
Tetyana Yakovenko, Ukraine
We’ve determinated the conditions under which trajectories of stochasticprocesses from Orlicz spaces of random variables belong with probability one tothe functional Sobolev spaces in the case of noncompact parametric space. Ob-tained results could be applied for the estimation of the rate of convergence forwavelet expansion in the norm of space Lq(R) for the stochastic processes fromLp(Ω) space.
Definition 1. Stochastic process X ∈ Lp(Ω) is called differentiable by thenorm in the space Lp(Ω) in the point t0 ∈ T , if there is such random variableX [1](t0) ∈ Lp(Ω) , that
limh→0
‚‚‚‚X(t0 + h)−X(t0)
h−X [1](t0)
‚‚‚‚Lp(Ω)
= 0.
The process X is differentiable by the norm in the space Lp(Ω) on T , if it isdifferentiable by the norm in every point of the parametric set. Then this processX [1] = X [1](t), t ∈ T is called derivative by the norm in Lp(Ω) of the processX .
Theorem 1 Lets consider numbers p and q , such that 1 < p ≤ q . If the stochas-tic process X = X(t), t ∈ R is separable, measurable, N times differentiableby the norm in Lp(Ω) and for all a, b ∈ R such that b − a ≤ 1 the followingconditions are fulfilled:
1) ∀k = 0, N supt∈[a,b)
“E˛X [k](t)
˛p”1/p
≤ 1(max|a|,|b|)τ , where
τ > 11−γ
“1 + 1
p− γ
q
”and 0 < γ < 1 – some fixed number;
2) sup |t− s| ≤ ht, s ∈ [a, b)
“E|X [N ](t)−X [N ](s)|p
”1/p
≤ Ca,bhα , where
α > 1p− 1
qand there exists 0 < c < ∞ such that
Ca,b ≤ c(b−a)α supa≤t<b
“E|X [N ](t)|p
”1/p
,
then X(t) ∈ WNq (R) with probability one.
Department of Probability Theory and Mathematical Statistics,Faculty of Mechanics and Mathematics,Kiev National Taras Shevchenko University,Volodymyrska 64, 01033, Kyiv, Ukrainee-mail: [email protected]
273
Some properties of ϕ-sub-Gaussian queues
Rostyslav Yamnenko, Ukraine
Consider a workload process of the general single server queue
A(t) = mt+ σ
NXi=1
ZHi (t), t ≥ 0,
where m > 0 is the mean input rate, σ > 0 is a variance coefficient and c > mis the constant output rate, A(t) is the amount of work which arrives to beprocessed in the interval [0, t) , ZHi – some independent input processes. Considerthe probability that the amount of surplus exceeds a certain level x (buffer size),i.e.
Q = P
supt>0
(A(t)− ct) > x
ff.
It is well-known [1] that for the model with the fractional Brownian motioninput process the next asymptotic holds true for large enough buffer size x
Q ∼ exp
−C
2H
2κ2H
x2−2H
ff,
where κH = HH(1−H)1−H .The properties of FBM make it natural choice in modeling traffic through
telecommunication networks since it exhibits long-range dependence and self-similarity. But since in most cases real processes are Gaussian only asymptoti-cally or not Gaussian at all there arises a problem of introduction of more generalclass of arrival processes than Gaussian one. From the such viewpoint the classesof ϕ -subgaussian and strictly ϕ -subgaussian random processes are of significantinterest as a natural expansion of the class of Gaussian random processes [2].
Here the model with strictly ϕ -subgaussian generalized fractional Brownianmotion (strictly ϕ -GFBM) input is presented. As a result we obtain the upperbound of the buffer overflow probability for such model.
Let (Ω,F,P) be a standard probability space.
Definition. We call the process ZH = (ZH(t), t ∈ T ) strictly ϕ -GFBM withHurst index H ∈ (0, 1) if ZH is strictly ϕ -sub-Gaussian process with stationaryincrements and covariance function
RH(t, s) = EZH(s)ZH(t) =1
2
“t2H + s2H − |s− t|2H
”.
274
Let C = c−mσC∆
, ε = xσC∆
, where C∆ is constant from the definition of spaceSSubϕ(Ω) (space of strictly ϕ -sub-Gaussian random variables) (see [2] for moredetails). Then
P
supt>0
(A(t)− ct > x)
ff= P
(supt>0
1
C∆
NXi=1
ZHi (t)− Ct
!> ε
)
and the following theorem takes place
Theorem 1 Let ZHi = (ZHi (t), t ≥ 0), i = 1, N be independent strictly ϕ -GFBMrandom processes from the class Ψq
x0 with identical Hurst parameter H ∈ [0.5, 1) ,
qH > 1 , ϕ(x) =
8<:xq
xq0, |x| > x0,
x2
x20, |x| ≤ x0.
Let C > 0 and γ > 1 be some constants.
Then for all ε ≥ ε0 the following inequality holds true
P
(supt>0
1
C∆
NXi=1
ZHi (t)− Ct
!> ε
)≤ L(γ)ε
q(1−H)(q−1)H exp
−κ(γ)ε
q(1−H)q−1
ff,
where
κ(γ) =x
qq−10 C
qHq−1 (q − 1)(γ − 1)
1q−1 (γqH − γ)
qH−1q−1
N1
q−1 (q − qH)q−qHq−1 (qH − 1)
qH−1q−1 (γqH − 1)
qHq−1
and L(γ, ε) is known constant that is uniformly bounded on ε for any γ > 1 .
[1] I. Norros. On the use of Fractional Brownian motions in the Theory of Con-nectionless Networks. IEEE Journal on selected areas in communications, Vol.13,No.6 pp. 953–962, 1995.
[2] V.V. Buldygin and Yu.V. Kozachenko. Metric Characterization of RandomVariables and Random Processes. American Mathematical Society, Providence,RI, 2000.
Kyiv National Taras Shevchenko University,Department of Probability Theory and Math. Statistics,Volodymyrska str., 64, 01033, Kyiv, Ukrainee-mail: [email protected]
275
On application of the INAR(m) model forconstructing of pseudorandom sequences
Andrei Yarmola, Belarus
Simulation of complex systems in economics, genetics needs pseudorandom se-quences generators (PRSG), also PRSGs are used in data security system. Theproblem of constructing pseudorandom sequences which are statistically to ran-dom uniformly distributed sequences is very topical. In this work we introduceda new method of constructing PRSG based on the INAR(m) model of integervalued time series [1].
The INAR(m) model was introduced by Alzaid and Al-Osh [1] as integervalued analogue of AR(m) model zt ∈ Z+ = 0, 1, 2, . . . , t = 1, 2, . . . :
zt =
mXi=1
pj zt−j + ηt, t = m+ 1,m+ 2, . . . , (1)
where is the operator of binomial thinning: pj zt−j =Pzt−j
i=1 ξ(j)t,i ;
ξ(j)t,i : t = m+1,m+2, . . . ; i = 1, 2, . . . ; j = 1, . . . ,m are i.i.d. random variables,Pξ(j)t,i = 1 = 1−Pξ(j)t,i = 0 = pj ; ηt ∈ Z+, t = m+1,m+2, . . . are randomvariables, such that ηt is not correlated with zt−1, . . . , zt−m , t = m+1,m+2, . . . ;z1, . . . , zm are some initial values.
We use the idea of binomial thinning for a new method of construction PRSG.The INAR-generator is defined as follows:
xt = (
mXi=1
θi
xt−iXj=0
ξ(i)
(t−1)N+j) mod N, t = m+ 1,m+ 2, . . . , (2)
where ξ(j)t ∈ 0, 1 : j = 1, . . . ,m are output sequences of some primary ("ba-sic") binary generators G1, . . . , Gm ; in general: θj ∈ AN = 0, . . . , N − 1 ,j = 1, . . . ,m , θm > 0 . Further we consider a particular case of the model (2)θj ∈ A2 = 0, 1 , j = 1, . . . ,m . The distinctive feature of the INAR generator isthe sum of random number of random variables
Pxt−i
j=0 ξ(i)
(t−1)N+j in (2).To analyze probabilistic properties of the INAR-generator (2) we assume that:
the output sequences ξ(1)t , . . . , ξ(2)t of "basic" generators be sequences of bi-nary random variables. Let r =
Pmj=1 θj ≥ 1 be Hamming weight of the binary
vector θ = (θ1, . . . , θm)′ .
Theorem 1 If ξ(j)t are i.i.d. binary random variables,Pξ(j)t = 1 = 1 − Pξ(j)t = 0 = pj , j = 1, . . . ,m , t = m + 1,m + 2, . . . , thenthe sequence xt generated by (2) is a homogeneous m -th order Markov chain.
276
Lemma 1 If r ≥ N − 1 and 0 < pj < 1 , j = 1, . . . ,m , then Markov chain xt isergodic.
Consider an important for practice case when N = 2 , and all "basic" genera-tors are of the same type:
pj =1
2(1 + ε), j = 1, . . . ,m, |ε| < 1, (3)
where ε is the value which define a "defect" of "basic" generators.To evaluate the deviation of the probability distribution of Markov chain
xt from the probability distribution of the uniformly distributed random se-quence we use the following functionals introduced in discriminant analysis ofMarkov chains [2]: the average deviation of transition probability distributionP = (pi0,...,im−1,im) , i0, . . . , im ∈ A2 from the uniform probability distribution
∆ = 2−mX
i0,...,im∈A2
|pi0,...,im − 0.5|;
the octahedral norm of the deviation of stationary m -dimensional probabilitydistribution Π∗ = (π∗i1,...,im) , i1, . . . , im ∈ A2 from the uniform probability dis-tribution
Zm =X
i1,...,im∈A2
|π∗i1,...,im − 2−m|.
Theorem 2 Under conditions (3) for the functionals ∆ , Zm it holds:
∆ = |ε/2|r(1 + |ε|)r, Zm ≤ Z+,
Z+ = (1−ε)˛ε
2
˛r(1+ |ε|)r−1
1 +
m−1Xj=1
22(rj−j)|ε|rj
(1 + |ε|)rj
jYk=1
(1 + (−ε)θk−j )
!+O(ε2r),
where rj =Pjk=1 θk , j = 1, . . . ,m− 1 .
References
[1] Alzaid, A.A. and Al-Osh, M. An integer-valued pth-order autoregressive struc-ture (INAR(p)) process. - "Journal of Applied Probability", 1990, 27, pp.314-324.
[2] Kharin Yu., Kostevich A. Discriminant analysis of stationary finite Markovchains. - "Math. Methods of Statistics", 2004, Vol. 13, No.1, pp. 235-252.
Faculty of Applied Mathematics and Informatics,Belarusian State University,4 Independence av., 220030 Minsk, Belaruse-mail: [email protected]
277
On stability of the functional differentialequations with Markov switchings
Yasinsky V.K., Yurchenko I.V., Ukraine
Let the stochastic process x(t) ≡ x(t, ω) ⊂ Rn is defined on the probabilisticspace (Ω, F, P ) with the stream of σ -algebras Ft, t ≥ t0 , Ft ⊂ F with the helpof the following functional differential equation
dx(t) = a(t, xt, ξ(t))dt+ b(t, xt, ξ(t))dw(t), t ≥ t0, (1)
with initial condition
x(t0 + θ) = ϕ(θ), θ ∈ [−τ, 0], τ > 0. (2)
Here xt ≡ x(t + θ), θ ∈ [−τ, 0] ; ξ(t) ≡ ξ(t, ω), t ≥ t0 ⊂ Rn is a stochasticcontinuous homogeneous Feller Markov process with right-continuous realizationson a compact phase space Y ; D is a Skorokhod space of the functions ϕ(θ) ⊂ Rn ,a : [t0,∞) × Cn([−τ, 0] × Y ) → Rn is a continuous mapping on its arguments;b : [t0,∞)× Cn([−τ, 0]× Y ) →Mn(Rn) is a matrix of n× n -dimension; w(t) isa n -dimensioned process of the Brownian motion.
n -dimensioned stochastic absolutely continuous on variable t ≥ t0 processx(t) will be named the solution of the problem (1), (2) on the set [t0, T ) ⊂ R+ ,if for all T1 ⊂ [t0, T ) , t ∈ [t0 − τ, T1) with probability 1 the following equality isvalid
x(t) =
8<:ϕ(t− t0), for t ∈ [t0 − τ, t0),
ϕ(0) +tRt0
a(s, xsξ(s))ds+tRt0
b(s, xs, ξ(s))dw(s), for t ∈ [t0, T1].
If arbitrary two solutions of (1), (2) are equal with probability 1 on the arbi-trary segment t ∈ [t0, T ) , then the solution of (1), (2) on this set will be consideredto be unique.
Let us denote x(t, t0, ϕ, y) the solution of (1), (2), where the conditionξ(t0) = y is valid. The part of solution trajectory on the time segment [t − τ, t]will be denoted xt(t0, ϕ, y) ≡ x(t+ θ, t0, ϕ, y), θ ∈ [−τ, 0] .
Let consider the scalar continuous functional on all variables
V : R+ × Cn([−τ, 0])× Y → R1.
Let the global Lipschitz condition
|V (t, ϕ, y)− V (t, ψ, y)| ≤ L‖ϕ− ψ‖278
is valid for all ϕ,ψ ∈ Cn([−τ, 0]) and
supt≥0
|V (t, 0, y)| = α <∞,
where ‖ϕ‖ = sup−τ≤θ≤0
|ϕ(θ)| .
Let us denote the linear operator
(T (t)V )(s, ϕ, y) ≡ZY
EV (s+ t, xs+t(s, ϕ, y), z)P (t, y, dz).
with the help of transition probability P (t, y, dz) of a Markov processξ(t) ⊂ Rn .
Theorem 1. Let the functional V is continuous. Then:1) result of action of the operator T (t) on V (t, ϕ, ξ) is continuous function
i.e. T : C(Y ) → C(Y ) , where Y ≡ [0,∞)× Cn([−τ, 0)× Y ) ;2) operator T (t), t ≥ 0 creates the semigroup:
T (t1 + t2) = T (t1) · T (t2), ∀ t1, t2 ≥ 0;
3) family of the linear operators on the phase space Y denotes the stochasticcontinuous Markov process with right-continuous realizations.
Theorem 2. Let:1) a(s, 0, y) ≡ 0, b(s, 0, y) ≡ 0 ;2) local Lipschitz condition is valid
|a(t, ϕ, y)− a(t, ψ, y)|+ ‖b(t, ϕ, y)− b(t, ψ, y)‖ ≤ Lr‖ϕ− ψ‖
for all t ≥ 0 , y ∈ Y , Lr, r > 0 and ϕ,ψ ∈ Ur(0) ≡ ϕ ∈ Cn([−τ, 0]) |, ‖ϕ‖ < r ;3) |a(t, ϕ, y)| + |b(t, ϕ, y)| ≤ K(‖ϕ‖ρ + α) , where K > 0 ,
‖ϕ− ψ‖ρ ≡0R−τ|ϕ(θ)− ψ(θ)|ρ(dθ) and ρ is some probabilistic measure;
4) functional V (s, ϕ, y) exists and the following condition is valid
c1|ϕ(0)|p1 ≤ V (s, ϕ, y) ≤ c2‖ϕ‖p2
for c1, c2 > 0 , p2 ≥ p1 > 0 and all s ∈ R+ , y ∈ Y and ϕ ∈ Cn([−τ, 0]) ;5) for some c3 > 0 and p ∈ (0, p1] the following condition is valid
(LV )(s, ϕ, y) ≤ −c3|ϕ(0)|p,
for all s ≥ 0 , y ∈ Y and ϕ ∈ Cn([−τ, 0]) ; L is a weak infinitesimal operator ofspecial form.
Then the trivial solution of the problem (1), (2) is asymptotical p -stable.
Chernivtsi National University named after Yurij Fedkovich,vul. Kotsiubinskogo, 2, Chernivtsi, 58012, Ukrainee-mail: [email protected]
279
The theorem of the regeneration typeYelejko Yaroslav, Zabolotskyy Taras, Ukraine
Let Gεij(dx) , ( i, j is natural numbers) - be a family of complex mea-sures on positive semi-line. Define V εij(dx) a variation of the measure Gεij(dx)and let Vε(dx) = (V εij(dx))∞i,j=1, Gε(dx) = (Gεij(dx))∞i,j=1 . Let us assumethat there exist natural number m such that, for all i 6= j , i ≥ m ,j ≥ m it holds that Gεij(dx) = 0 . Denote the matrix of regeneration of
Gε(dx) by Hε(dx) =∞Pk=0
Gk∗ε (dx) , where G0∗ε ([0, x]) = I , (x ≥ 0), and
G(k+1)∗ε ([0, x]) = Gk∗ε ∗Gε([0, x]) =
xR0
Gk∗ε ([0, x− y])Gε(dy) .
Let us assume that the following conditions hold
limε→0
Gεij(dx) = Gij(dx), Gij(dx) = 0 (i 6= j), Gii(dx) ≥ 0, Gii([0,∞)) = 1; (1)
limε→0
1
ε(I − V ε([0,∞))) = D = dij∞i,j=1, dij <∞; (2)
limε→0
1
ε(I −Gε([0,∞))) = C = cij∞i,j=1, cij <∞; (3)
limT→∞
supε>0
∞ZT
xV εij(dx) = 0. (4)
Define mi =∞R0
xGii(dx) , M = diagm1,m2, ... . Condition (4) implies that
for all natural numbers i the inequality 0 < mi <∞ holds. Assume that
infimi > 0, sup
imi <∞. (5)
Definition 1 A family of functions fε(x) defined for x ≥ 0 is called uniformlydirectly integrable by Riman if the following conditions hold sup
ε>0supx≥0
|fε(x)| <∞ ,
limm→∞
supε>0
∞Pn=m
supn≤x≤n+1
|fε(x)| = 0 and limδ→0
δ supε>0
∞Pn=0
( supnδ≤x≤(n+1)δ
fε(x)−
infnδ≤x≤(n+1)δ
fε(x)) = 0 .
Theorem 1 Let vector function−→f ε(x) is uniformly directly integrable by Riman
and limε→0
∞R0
−→f ε(x)dx =
−→b , for the measures Gεij(dx) conditions (1)-(5) hold and
matrix D is positive defined, matrix (Φ(λ) − I) , where Φ(λ) =∞R0
eiλxG(dx) , is
invertible, then for every s ≥ 0 it holds−→f ε ∗Hε(x) → e−tM
−1CM−1−→b ,
with ε→ 0 , x→∞ and εx→ t .
Lviv Ivan Franko National University
280
Transport Combinatorial Stochastic Tasks andTheir Solving
O.O. Yemets’, T.O. Parfonova, Ukraine
Possible combinatorial conditions, which can be imposed on the permissiblesolving, and possible probable parameters of a task are not taken into accountin classical transport tasks. More adequate modeling of transport optimizationproblems demands taking into consideration these aspects [1-9].
Let’s consider Task: find volumes xij of conveyances from the supplier i tothe consumer j for all i ∈ Jm , j ∈ Jr (hereinafter Jn = 1, 2, ..., n - the setof the first n natural numbers) that provide minimal total cost of conveyancesof a homogeneous product between the manufacturers m and the consumers n ,if the cost cij of conveyances of the unit of output from the manufacturer i tothe consumer j is known The maximum possible volumes of production in thepoint i equals ai , where i ∈ Jm . The minimum possible volumes of consumptionin the destination j are given and equals according to bj , ∀j ∈ Jn . Magnitudesai will be considered as random numbers with known mathematical expectationsM (ai) .
It is considered that the transportation of the cargo is possible to be carriedout in the certain capacities in the quantity k of volumes g1, g2, ..., gk accordingly.
The model of such Task can be submitted as: find(1) F (x∗) = min
x∈Rk
mPi=1
nPj=1
cij · xij ;
(2) x∗ = argminx∈Rk
mPi=1
nPj=1
cij xij
under conditions(3)
mPi=1
xij ≥ bj ,∀j ∈ Jn;
(4)nPj=1
xij ≤ ai,∀i ∈ Jm;
(5) xij ≥ 0,∀i ∈ Jm,∀j ∈ Jn;(6) x = (x11, ..., x1n, ..., xi1, ..., xin, ..., xm1, ..., xmn) ∈ Ekν (G) ,
where m, n – are given constants, k = mn , ai , bj , cij are given (for all possibleindexes) real positive numbers, Ekν (G) is the set of permutations of k elementsof the given multiset of possible volumes of conveyances G = g1, ..., gk , amongwhich there is ν different real numbers.
Task (1) - (5) is classical transport Task. Task (1) - (6) will be entitled astransport combinatorial task on permutations with stochastic parameters.
As is well known, one of the approaches of the solving of tasks with stochas-tic parameters consists of these parameters replacement by their mathematicalexpectations. Then in model (1) - (6) the condition (4) will be transformed into
(7)nPj=1
xij ≤M(ai);∀i ∈ Jm.
Task (1) - (3), (5) - (7) is the determined task of Euclidean combinatorialoptimization with linear criterion function and linear restrictions on permutations.
281
It can be solved by a method of combinatorial cutting off, which was offered andproved in [2].
The solution of Task (1) - (3), (5) - (7) cannot satisfy the condition (4) atthe certain realization of random magnitudes ai . The risk of infringement ofconditions of Task (1) - (6) can be taken into account in the following way. Let’sdesignate xM the solution of Task (1) - (3), (5) - (7), and consider function,which estimates risk of infringement of conditions of Task and increase of criterionfunction at transition from the solution xM to the solution x of Task (1) - (6):mPi=1
δAi (ai−nPj=1
xij) +nPj=1
δBj (bj−mPi=1
xij)+δC
mPi=1
nPj=1
cij(xij−xMij )+mPi=1
∆Ai (M(ai)−ai).
In last formula variables δAi , δBj , δC , ∆Ai express a payment for infringement
of the appropriate conditions.
1. Y.G. Stoyan, O.O. Yemets’. The Theory and the Methods of Euclideancombinatorial optimization. – K.: ISIE, 1993. – 188 pp. (in Ukrainian).
2. Y.G. Stoyan, O.O. Yemets’, E.M. Yemets’ Oprtimization on the pol-yarrangements: theory and methods. - Poltava: EPC PUCC, 2005. – 103 pp.
3.Valuiskaya O.A., Emets O.A. (Yemets’ O.O.), Romanova N.G. Stoyan-Yakovlev‘s modified method applied to convex continuation of polynomials definedon polypermutations // Computational Mathematics and Mathematical Physics.- 2002. - V. 42. - No. 42. - P. 566 - 570.
4. Emets O. A. (Yemets’ O.O.), Kolechkina L. N. Solution of OptimizationProblems with Fractional-Linear Objective Functions and Additional Linear Con-straints on Permutations // Cybernetics and Systems Analysis. - 2004 - V. 40, 3, - P. 329 - 339.
5. Barbolina T.N., Emets O.A. (Yemets’ O.O.) An all-integer cutting methodfor linear constrained optimization problems on arrangements // ComputationalMathematics and Mathematical Physics. - 2005. - V. 45. - No. 5. - P. 254 - 261.
6. Yemets O.A. (Yemets’ O.O.) The optimization of linear and convex func-tions on a Euclidean combinatorial set of polypermutations //Comp. Maths.Math. Phys. - 1994. - V. 34, N6. - P. 737-748.
7. Emets O.A. (Yemets’ O.O.) Extremal properties of nondifferentiable con-vex functions on Euclidean sets of combinations with repetitions // UkrainianMathematical Journal. - 1994. - V.46, N6. P. 735-747.
8. Emets O.A. (Yemets’ O.O.), Roskladka A.A. On estimates of minima crite-rion functions in optimization on combinations // Ukrainian Mathematical Jour-nal. -1999. - Vol. 51. - No 8. - P. 1262-1265.
9. Emets O.A. (Yemets’ O.O.), Kolechkina L.M. Optimization problem onpermutations with linear-fractional objective function: properties of the set ofadmissible solutions // Ukrainian Mathematical Journal. -2000. - Vol. 52. - No12. - P. 1858-1871.
Poltava University of Consumer Cooperatives in Ukraine,P. B. 1671, Poltava, 36003, Ukrainee-mail: [email protected]
282
Convergence of solutions of backwardstochastic differential equations
Irina Yerisova, Ukraine
We consider the limit behavior of solutions of backward stochastic differentialequations, as ε→ 0
Y εt = gε(XεT ) +
TZt
fε(s,Xεs , Y
εs ) ds −
TZt
Zεs dBs , 0 ≤ t ≤ T (1)
where (Y εt , Zεt ), 0 ≤ t ≤ T — the unique FBt −adapted Rk × Rk×d−valued
solution of equation (1) and process Zεt such that ETR0
||Zt||2 dt < ∞ .
We assume that the coefficients gε and fε —k−valued vectors for which someconditions hold.
In the equation (1) Xεt , t ≥ 0 — Rd -valued diffusion process which is the
strong solution of the stochastic differential equation
Xεt = xε +
tZ0
bε(s,Xεs ) ds +
tZ0
σε(s,Xεs ) dBs , 0 ≤ t , (2)
Under some conditions for the coefficients of this equation Xε =⇒ X , as ε→ 0 .Let (Yt, Zt), 0 ≤ t ≤ T — the unique FWt −adapted Rk × Rk×d−valued
process which solves the backward stochastic differential equation
Yt = g(XT ) +
TZt
f(s,Xs, Ys) ds −TZt
Zs dWs. , (3)
In the report we prove Y ε ⇒ Y in the sense of Meyer-Zheng, az ε→ 0 .
Institute of Applied Mathematics and Mechanicsof the National Academy of Sciences of Ukraine,Donetsk, 83114, st. R.Luxemburg, 74e-mail: [email protected]
283
On a random system of nonlinear functionaldifferential equations with nonlinear mixed
random maximaYuldashev T.K., Kyrgyzstan
In last 25-30 years a new special class of functional differential equations (FDE)with deviation arguments has been studied. The right-hand side of FDE along withthe simple argument "t" also depends on the construction
maxx(τ)|τ ∈ [δ1(t); δ2(t)], 0 ≤ δ1(t) ≤ δ2(t) <∞.Such classes of FDE we call equations with maxima. FDE with maxima have beenconsidered by many authors, for example see [1-8]. In applications, the maximumarises when the control law corresponds to the maximal deviation of the regulatedquantity. If the control law takes into account also the maximal velocity of devia-tion of this quantity, then the process is governed by equations with maxima [1,5].
The aim of this paper is to study the sufficient conditions of existence anduniqueness of the random solution of the initial value problem for nonlinear ran-dom FDE with mixed random maxima of the first order. Here we use the methodof successive approximations in combination with the method of compressing map-ping. We obsere that our present work is a further development of the theory ofnonlinear FDE with maxima.
We consider a system of nonlinear random FDE of the formx′(t) = F (t, x(t),maxx(τ)|τ ∈ [δ1/δ2], u(t)) , t ∈ T0
with initial condition x(0) = x0 = const < ∞, whereF (t, x, y, u) ∈ C(D), D ≡ T0 ×X ×X ×X, T0 ≡ [0;T ], 0 < T <∞, X ⊂ Rn isbounded closed set, δi = δi(t, x(t), ϑ(t)) are random deviations of mixed type,i = 1, 2, [δ1/δ2] ≡ [minδ1; δ2;maxδ1; δ2], 0 ≤ δi(t, x, ϑ) ≤ T , functionsu(t), ϑ(t) ∈ X are standard Winer processes.
We note that FDE with mixed maxima have not sean considered by any au-thors. The solutions of FDE with mixed maxima have singularities.
1. Magomedov A.R. Ordinary Differential Equations with maxima. - Bakou: Elm, 1991.- 220 p. (Russian)2. Muntyan V.I., Shpakovich V.P. To Question about Continuous Dependence ofSolutions of Differential Equations With maxima from Parameter // Questions ofStability of Integral Variety in Equations of Math. Phisics. - Kiev: Nauk. Dumka, 1987.-P. 49-55. (Russian)3. Petuhov V.R. Questions of Qualiutive Investigation of Solutions of Equations withmaxima // Izv. vuzov. Math., 1964, 3. -P. 116-119. (Russian)4. Voulov H.D. , Bainov D.D. On the Asymptotic Stability of Differential Equationswith maxima // Rend. Circ. Math. Palermo. Ses. 2., 1991 (40), 3. -P. 385-420.5. Bainov D.D., Milisheva S.D. Justification of the Averaging Method for functionaldifferential equations with maxima // Nouvelle Serie, Tome 38 (52), 1984. -P. 149-152.
Kyrgyz state law academye-mail: [email protected]
284
Random hypermeasures and stochasticintegration
Andriy Yurachkivsky, Ukraine
Let X be a σ -algebra of subsets of a set X . We call a random function µ on Xa stochastic integrator if it is additive and for any decreasing sequence (An) ∈ XN
with empty intersection µ(An)P−→ 0 .
Fixing one variable – A ∈ X or ω ∈ Ω – prior to another, we may regardstochastic integrator either as a map from the space of R -valued random vari-ables or as a random element taking values in some topological space. Within theframework of the first approach the integral
Rfdµ was defined by Turpin and
studied by Radchenko. To develop the second approach one has to construct atopological space which almost all realizations of stochastic integrator belong to.
Let (X, ρ) be a metric space and X be its Borel σ -algebra. Denote
Φ = ϕ ∈ C(X) : ∀ x, y ∈ X |ϕ(x)| ≤ 1 & |ϕ(x)− ϕ(y)| ≤ ρ(x, y),
Q – the set of all signed measures on X endowed with the norm‖q‖ = sup
ϕ∈Φ|Rfdq| .
Theorem 1 Let (X, ρ) be a Polish space of infinite cardinality such that the classΦ separates its points. Then the normed space Q is incomplete and its completiondoes not depend, up to linear homeomorphism, on ρ .
We call the completion elements hypermeasures. They can be regarded, simi-larly to measures, as functionals – but on bounded Lipschitz functions only.
Theorem 2 Let X be a bounded Polish space, µ be a stochastic integrator onX and let there exist an exhaustive sequence (Dn) of measurable partitions of Xsuch that each atom of Dn is the join of two atoms of Dn+1 and
supn
24E
XD∈Dn
µ(D)2!1/2
+ 2n maxD∈Dn
diamD
35 <∞.
Then there exists a random hypermeasure on X whose value on any boundedLipshitz function ϕ almost surely equals
Rϕdµ .
This theorem allows to define the integralR t0
RΘϕ(u, θ)µ(dudθ) , where µ
is a generalization of martingale measure (strong orthogonality of µ(·, A1) andµ(·, A2) for disjoint A1 and A2 is not required) and to prove its properties simi-lar to those of integral by martingale measure.
Department of Mathematics and Theoretical Radiophysics,Kyiv University,vul. Volodymyrska 64, Kyiv, 01033, Ukrainee-mail: [email protected]
285
On the diffusion process with the driftcontaining delta-function concentrated on a
circle
Ludmila Zaitseva, Ukraine
We construct the generalized diffusion processes with the drift, containingdelta-function concentrated on a circle, as the strong solution to a stochasticdifferential equation.
For a given parameter a > 0 we consider the circle S = x ∈ <2 : ‖x‖ = a.Let the pair of independent Wiener processes (wt)t≥0 and ( bwt)t≥0 , the numberq ∈ [−1, 1] and the function α : S → Tx(S) (Tx(S) denotes the tangent space forS at the point x ∈ S ) be given. We investigate the following system of stochasticdifferential equations in <28><>:
drt = qdηt +1
2rtdt+ dwt,
dθt = α(θt)dηt +1
rtd bwt, (1)
where (rt, θt)t≥0 is the polar coordinates of the unknown process (xt)t≥0 in <2,(ηt)t≥0 is the local time of the process (rt)t≥0 at the point a.
We show that under the standard assumptions on the coefficient α(·) thestrong solution to the system (1) exists and is unique. We prove that the process(xt)t≥0 has a modification that is jointly measurable w.r.t. starting point and theWiener noise. The Markov property of the process (xt)t≥0 is also proved.
Theorem 1 The process (xt)t≥0 is the generalized diffusion process in the senseof Portenko N.I. (see [1]). The generalized drift vector of this process is equal to(qν(x) + α(x))δS(x) and the diffusion matrix is the identity matrix. Here ν(x) isthe normal vector to the circle S at the point x ∈ S, δS(·) is the surface measureon S.
References
[1] N.I. Portenko Generalized diffusion processes // Providence, Rhode Island.– 1972.
Kyiv Taras Shevchenko university,Volodymyrska 64, Kyiv, 01033, Ukrainee-mail: [email protected]
286
International experience in actuarial education
N.Zinchenko, Ukraine
We discuss the system of actuarial education and certification in Scandinavia,Central and West Europe as well as general requirement of IAA and Groupe Con-sultatif. The main attention is focused on develop the curriculum for postgraduateeducation of actuaries, the description of curriculum development according to theCore Syllabus of the Groupe Consultatif and modules prepared by European Ac-tuarial Academy and their comparison with actuarial courses developed in KyivNational Taras Shevchenko University.
The work is made under the project IB-JEP-25054-2004.
Kyiv National Taras Shevchenko University,Department of Mechanics and Mathematics,Volodimirska, 64, 01033, Kyiv, Ukrainee-mail: [email protected]
Ruin probabilities for the Cramer-Lundbergmodel with stochastic premiums
N.Zinchenko, A.Andrusiv, Ukraine
The simple diffusion approximation and de Vylder approximation for the ruinprobability for the Cramer-Lundberg model with stochastic premiums are in-troduced and investigated similarly to the case of classic Cramer-Lundberg riskmodel.The characteristic equations for the Lundberg coefficient are presented forcertain classes of stochastic premiums and claims, the variant of the strong in-variance principle for the risk process is discussed. The case of Sparre Andersenrenewal model is also investigated.
Kyiv National Taras Shevchenko University,Department of Mechanics and Mathematics,Volodimirska, 64, 01033, Kyiv, Ukrainee-mail: [email protected]
287
Comparison of Bayesian and modifiedcompromise estimators by use of criterion of
relative modulus risk
Igor V. Zolotukhin, Russian Federation
This presentation deals with the problem of estimation of normal distribu-tion variance σ2 , when the distribution parameters are unknown, but a prioriinformation is present.
In the typical situation one can suppose that expectation a and a varianceσ2 of a priori estimate just as this a priori estimate σ2
apr are known. By use ofthe conjugate family of inverse gamma distribution with parameters defined by aand σ2 , Bayesian estimator minimizing relative modulus risk can be constructed.It is possible to show that the risk function of Bayesian estimator is unbounded,
but at the same time the standard estimate S2 = 1n−1
nXi=1
(xi − x)2 has constant
risk.The following modified compromise estimator σ2
MC is suggested. It coincideswith boundary of the confidence interval (constructed by the standard estimateS2 ) if σ2
apr falling outside the confidence interval, and coincides with the a prioriestimate if σ2
apr falling inside the confidence interval, i.e.:
σ2MC =
8<:γ21S
2 if σ2apr < γ2
1S2,
σ2apr if γ2
1S2 < σ2
apr < γ22S
2,γ22S
2 if γ22S
2 < σ2apr.
The distribution of suggested estimator is found, and the relative modulusrisk is defined. The algorithm for searching of the best values γ2
1 and γ22 , which
minimized the risk value is suggested too. This algorithm is implemented by thesoftware package Mathematica. It is also shown that for the best values γ2
1 and γ22
the risk of the suggested estimator is very close to the risk of Bayesian estimator,and at the same time the risk function of this estimator σ2
MC is bounded.
St.-Petersburg Branch of Institute of Oceanology,Russian Academy of Sciences,198255 St.Petersburg, Leni Golikova 29/5-77e-mail: [email protected]
288
On local linear estimation in nonparametricerrors-in-variables models
Silvelyn Zwanzig
Consider observations (x1, y1), ...(xi, yi), ...(xn, yn) , generated independently by
yi = g (ξi) + εi
xi = ξi + δi.
We assume a nonparametric model class for the regression function g , uniformlydistributed design points ξi , normal errors δi in the errors-in-variables equationand a known variance quotient k.
In [2] Fan and Troung (1993) supposed a kernel regression estimator for g andderived the optimal consistency rates for an asymptotic approach with increasingsample size. The kernel estimation method is based on deconvolution kernels. In[1] Fan and Gjibels (1996) described the idea of local polynomial estimation in anordinary regression model, where the design points ξi ∈ [0, 1] are known. Stauden-meyer and Ruppert (2004) have applied the idea of local polynomial estimation tothe errors-in-variables model. Their estimator is based on ordinary naive kernels:The asymptotic results are given for decreasing measurement error variances.
In this talk the local linear approach is considered with two different linearestimation methods, the naive estimation and the total least squares. The localenvironment is defined by deconvolution kernels. All asymptotic results are givenfor increasing sample sizes.
The methods are defined as follows. For each point x a local linear approxi-mation t(ξ) = β0 + β1(ξ− x) for the regression function g(ξ) is considered. Thenthe local naive estimator is given by bgnaive(x) = btnaive(0) = bβ0,naive, where
bβ0,naive = arg minβ0,β1
nXi=1
w∗i (x) (yi − t(xi))2 .
The local total least squares estimator is defined asbgtls(x) = bttls(0) = bβ0,tls, where
bβ0,tls = arg minβ0,β1
nXi=1
w∗i (x) minξ
ˆ(yi − t(ξ))2 + k (xi − ξ)2
˜.
In ordinary local linear regression common weights are wi (x) = K“ξi−xh
”. In
errors-in-variables models they cannot be used because of the unknown designpoints ξi. Instead we have chosen
w∗i (x) = K∗h
“xi − x
h
”,with Exi/ξi
K∗h
“xi − x
h
”= K
„ξi − x
h
«.
289
K∗ is the deconvolution kernel of K.The local naive estimator works well. It can be shown, that bgnaive(x) is a
consistent estimator of g(x) , especially that bβ1,naive is a consistent estimator ofg′(x).
The local total least squares estimator does not work well. It can be shownthat the estimator bβ1,tls is not a consistent estimator g′(x).
References
[1] J. Fan and I. Gjibels (1996) Local Polynomial Modelling and Its Application,London: Chapman and Hall
[2] J. Fan and Y. Truong (1993) Nonparametric regression with errors in variables.Ann. Statist. 21. 1900-1925
[3] J. Staudenmayer and D. Ruppert (2004) Local polynomial regression and sim-ulation extrapolation. J.R.Soc.B. 66, Part 1,pp 17 -30
[4] S. Zwanzig (2006) On local linear estimation in nonparametric errors-in-variables models. (2006) Technical report. Uppsala University
Uppsala University,Box 480, SE- 75106 Uppsala, SwedenE-mail: [email protected]
290
Author Index
Abushov, Q.U., 100Agaeva, C.A., 98, 100Aissani, D., 196Aliev, S.A., 101Aliyeva, T.A., 198Androshchuk, M., 102Androshchuk, T., 103, 104, 190Andrusiv, A., 287Angulo, J.M., 230Anh, V.V., 230Apostolov, S., 264
Barbulescu, A., 105, 185Baranovskyi, O., 224Belopolskaya, Ya., 107Bender, C., 109Bondarenko, G., 110Bondarev, B.V., 14, 111Borysenko, O., 112Borysenko, O.D., 112Boyaryshcheva, T., 80Bradul, N., 15Bratyk, M., 16Budkov, D., 113Buldygin, V.V., 18, 19, 114Bulinski, A., 116
Chabanyuk, Ya., 118Chani, A., 92Chernecky, V.A., 120, 122
Dikarev, V., 22Dorogovtsev, A.A., 124Doukhan, P., 125Dovgay, B., 30Drozdenko, M., 244
Elejko, T.Y., 127
Fedoryanich, T., 88Feshchenko, O., 73Filonenko, D., 128Finkelshtein, D., 129
Fomina, T., 90
Georgieva, O., 48Gerasin, S., 23Goncharenko, Ya., V., 25Gora, M., 27Gorodnii, M., 131Grygorjeva, I., 28Gusak, D., 132Gushchin, A., 134
Hajiyev, V.H., 136Hasratova, M.H., 137Huran, S., 138Huryn, A., 139
Ibayev, E.A., 198Iksanov, A., 34, 140Ilchenko, A. V., 142Ilchenko, S., 60Illicheva, L., 36Indlekofer, K.-H., 143Inoue, A., 144Ivanenko, D., 146Ivanov, A.V., 147
Kadankov, V.F., 149Kadankova, T.V., 151Karnaukh, E., 153Kartashov, M.V., 155Kartashov, Yu., 157Kartashova, S., 38Kechedzhy, K., 264Kharin, A., 158Kharin, Yu., 159Khasmiskii, R., 160Kirichenko, L., 161Klesov, O.I., 114, 143Klykavka, B., 162Knopova, V., 163Kolodii, A.M., 164Kolodii, N.A., 165Kolomiets, Yu.V., 167
291
Kondratiev, Yu., 129Kononchuk, P.P., 168Konstantinides, D.G., 169Konstantinov, O., 170Kopytko, B., 222Korkhin, A., 171Kotova, O.V., 40Kovalenko, V.M., 41Kozachenko, Yu.V., 173, 174Krasnitsky, S., 176Krenevich, A., 43Krykun, I., 177Kukush, A., 178Kulik, A., 157Kulinich, G., 44Kurchenko, O., 45Kutovyy, O., 178Kyurchev, D., 46, 48
Lebedev, E.A., 179Leonenko, N., 181Lepeyev, A.N., 182Lodatko, A., 50Lukashevich, P., 184Lytvynov, E., 129
Maftei, C., 185Maiboroda, R., 186Makhno, S., 187Makushenko, I.A., 179Malenko, A., 188Malyarenko, A., 245Masol, V., 51, 52, 54–56Masol, V.V., 245Masyutka, A., 193Matsak, I., 59Matusov, Yu.P., 58Mayzelis, Z., 264Melnyk, S., 264Mishura, Yu., 60, 62, 178, 189, 190Mitin, D.Yu., 191Moklyachuk, M., 193Moldavskaya, E., 195Mouhoubi, Z., 196Mykytyuk, I.O., 25
Nagornyi, V., 21Nakonechny, A.N., 127Nakonechny, O., 64Nasirova, T.H., 200Nasirova, T.I., 198Nazarenko, N.A., 191Negadaylov, P., 34Nikiforov, R., 65Norkin, B., 201Novak, S., 202Novosjadlo, A.F., 203
Ochkur, O., 44Olenko, A., 162, 204Orlovsky, I.V., 147Orsingher, E., 205
Palamarchuk, O., 207Panfilova, G.B., 18Parfonova, T.O., 281Pashko, A., 66, 67Pavlenko, T., 209Perestyuk, M.M., 173Perun, Yu., 220Petrenko, S., 227Pogany, T.K., 211Pogorilyak, O., 68Pokas’, S.M., 122Pokhylko, D., 213Polekha, M.Ya., 215Poloskov, I.E., 217Polotskyy, S., 70Ponomarenko, O., 71, 219, 220Popereshnyak, S., 52Portenko, M., 222Posashkov, S., 223Prabhu, N.U., 169Pratsiovytyi, M., 72, 73, 224
Radchenko, V., 225Rakhimov, G.M., 226Rakhimova, D.G., 226Rebenko, O., 227Romashova, L., 54Roskladka, A., 228
292
Rozora, I., 174Ruiz-Medina, M.D., 230Ryazantseva, V., 75Ryzhov, A., 230Rzhevska, A., 232
Sadiqova, R.H., 200Sakhno, L., 234Samoilenko, I, 235Seleznjev, O., 240Semenovska, N., 77Senko, I., 236Shalabh, 237Shatashvili, A.A., 93Shatashvili, A.D., 90, 93Shchestyuk, N., 97Shelyazhenko, P., 62Shervashidze, T., 95Shevchenko, G., 178, 238Shklovets, A., 161Shlyk, P., 158Shurenkov, G., 239Shykula, M., 240Sichkar, I., 86Sidelnyk, O., 241Silvestrov, D., 243–245Skirta, E., 246, 248Slobodyan, M., 55Slobodyan, S., 56Slyusarchuk, P., 80Slyvka, G., 78Smolyakov, A.I., 111Soloveyko, O., 82Sottinen, T., 250Stanzhytsky, O., 83Steinebach, J.G., 114, 251Stepanov, E.V., 111Stroyev, O.M., 155Struckmeier, S., 253Sugakova, O., 254
Tegza, A., 78Terdik, Gy., 256Teugels, J., 245Thomas, S., 258
Torbin, G.M., 84Tovmachenko, N., 86Tsapovs’ka, Z., 259Turchyn, V.N., 260Turchyn, Ye., 174Turchyna, N., 261Tymko, A., 263Tymoshenko, O.A., 19
Usatenko,O., 264Usoltseva, O., 265Ustimenko, V., 266
Vasiljev, S., 20Vasylenko, N., 268Vasylyk, O.I., 173Veretennikov, A., 269Veryovkina, G., 21Vovk, O., 22Vyzhva, Z., 271
Yakovenko, T., 273Yamnenko, R., 274Yampol’skii, V., 264Yarmola, A., 276Yasinsky, V.K., 278Yelejko, Ya., 280Yemets’, O.O., 281Yerisova, I., 283Yershov, A., 37Yuldashev T.K., 284Yurachkivsky, A., 285Yurchenko, I.V., 278
Zabolotskyy, T., 280Zabrovets, M.A., 58Zaitseva, L., 286Zalesky, B., 184Zelepugina, I., 67Zhmyhova, T.V., 14Zinchenko, N., 287Zolotukhin, I.V., 288Zrazhevsky, O., 32Zwanzig, S., 289
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