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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 357869 11 pageshttpdxdoiorg1011552013357869
Research ArticleStochastic Differential Equations withMulti-Markovian Switching
Meng Liu1 and Ke Wang2
1 School of Mathematical Science Huaiyin Normal University Huairsquoan 223300 China2Department of Mathematics Harbin Institute of Technology Weihai 264209 China
Correspondence should be addressed to Meng Liu liumeng0557sinacom
Received 31 August 2012 Revised 2 March 2013 Accepted 5 March 2013
Academic Editor Jose L Gracia
Copyright copy 2013 M Liu and K WangThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper is concerned with stochastic differential equations (SDEs) with multi-Markovian switching The existence anduniqueness of solution are investigated and the pth moment of the solution is estimated The classical theory of SDEs with singleMarkovian switching is extended
1 Introduction
Stochastic modeling has played an important role in manybranches of industry and science SDEs with single continu-ous-time Markovian chain have been used to model manypractical systems where they may experience abrupt changesin their parameters and structure caused by phenomenasuch as abrupt environment disturbances SDEs with singleMarkovian switching can be denoted by
where 120574(119905) is a right-continuous homogenous Markovianchain on the probability space taking values in a finite statespace S = 1 2 119873 and is F
119905-adapted but independent
of the Brownian motion 119861(119905) and
119891 R119899times R+times S 997888rarr R
119899
119892 R119899times R+times S 997888rarr R
119899+119898
(2)
Owing to their theoretical and practical significance (1)has received great attention and has been recently studiedextensively andwe heremention Skorokhod [1] andMao andYuan [2] among many others
However in the real world the condition that coefficients119891 and 119892 in (1) are perturbed by the same Markovian chainis too restrictive For example in the classical Black-Scholesmodel the asset price is given by a geometric Brownianmotion
where 120583 is the rate of the return of the underlying assert ]is the volatility and 119861(119905) is a scalar Brownian motion Sincethere is strong evidence to indicate that 120583 is not a constant butis a Markovian jump process (see eg [3 4]) many authorsproposed the following model
However many stochastic factors that affect 120583 are differentfrom those that affect ] Then the following model is moreappropriate than model (105) to describe this problem
where 120574119894(119905) is a right-continuous homogenous Markovian
chain taking values in a finite state space 119894 = 1 2 Anotherexample is the stochastic Lotka-Volterra model with singleMarkovian switching which has received great attentionand has been studied extensively recently (see eg [5ndash12])
2 Journal of Applied Mathematics
For the sake of convenience we take the following two-dimensional competitive model as an example
where 119909119894is the size of 119894th species at time 119905 119903
1198940(119895) represents
the growth rate of 119894th species in regime 119895 for 119894 = 1 2119895 isin S and 119861
1and 119861
2are independent standard Brownian
motions However there are many stochastic factors thataffect some coefficients intensely but have little impact onother coefficients in (6) For example suppose that thestochastic factor is rain falls and 119909
1is able to endure a damp
weather while 1199092is fond of a dry environment then the rain
falls will affect 1199092intensely but have little impact on 119909
Thus the above examples show that the study of thefollowing SDEs with multi-Markovian switchings is essentialand is of great importance fromboth theoretical and practicalpoints
switching among each other according to the movement ofthe Markovian chains It is important for us to discover the
properties of the system (8) and to find out whether thepresence of two Markovian switchings affects some knownresults The first step and the foundation of those studies areto establish the theorems for the existence and uniqueness ofthe solution to system (8) So in this paper we will give sometheorems for the existence and uniqueness of the solution tosystem (8) and study some properties of this solution Thetheory developed in this paper is the foundation for furtherstudy and can be applied in many different and complicatedsituations and hence the importance of the results in thispaper is clear
It should be pointed out that the theory developed in thispaper can be generalized to cope with the more general SDEswith more Markovian chains
The reason we concentrate on (8) rather than (11) is to avoidthe notations becoming too complicated Once the theorydeveloped in this paper is established the reader should beable to cope with the more general (11) without any difficulty
The remaining part of this paper is as follows In Section 2the sufficient criteria for existence anduniqueness of solutionlocal solution andmaximal local solution will be establishedrespectively In Section 3 the 119871
119875-estimates of the solutionwill be given In Section 4 we will introduce an example toillustrate our main result Finally we will close the paper withconclusions in Section 5
2 SDEs with Markovian Chains
Throughout this paper let (ΩF F119905119905isin119877+
P) be a completeprobability space Let 119861(119905) = (119861
1(119905) 119861
119898(119905))119879 be an 119898-
dimensional Brownian motion defined on the probabilityspace
In this section we will consider (8) Let 120574(119905) = (1205741(119905)
1205742(119905)) We impose a hypothesis(H1) 120574
1(119905) is independent of 120574
2(119905)
Then 120574(119905) is a homogenous vector Markovian chain withtransition probabilities
This implies minus1199021198941119894211989411198942
le lim inf119905rarr0
(120601(119905)119905) Thus
minus1199021198941119894211989411198942
= lim119905rarr0
120601 (119905)
119905 (24)
Using the definition of 120601(119905) gives
lim119905rarr0
1 minus 1198751198941119894211989411198942
(119905)
119905= lim119905rarr0
1 minus exp 120601 (119905)
120601 (119905)
120601 (119905)
119905= minus1199021198941119894211989411198942
(25)
which is the required assertion
4 Journal of Applied Mathematics
Lemma 4 Under Assumption (H2) for (1198941 1198942) (1198951 1198952) isin
S (1198941 1198942) = (1198951 1198952)
1199021198941119894211989511198952
= 1198751015840
1198941119894211989511198952
(0) = lim119905rarr0
1198751198941119894211989511198952
(119905)
119905(26)
exists and is finite
Proof By (H2) we note that for all 0 lt 120576 lt 13 exist0 lt 120575 lt 1such that
1198751198941119894211989411198942
(119905) gt 1 minus 120576 1198751198951119895211989511198952
(119905) gt 1 minus 120576
1198751198951119895211989411198942
(119905) lt 120576
(27)
provided 0 lt 119905 le 120575For forall0 le ℎ lt 119905 set 119899 = ⟨119905ℎ⟩ where ⟨119886⟩ = max
119899le119886119899 isin
Z Let
119875(11989511198952)
1198941119894211989611198962
(ℎ) = 1198751198941119894211989611198962
(ℎ)
119875(11989511198952)
1198941119894211989611198962
(119898ℎ)
= sum
(11990311199032) = (1198951 1198952)
119875(11989511198952)
1198941119894211990311199032
((119898 minus 1) ℎ) 1198751199031119903211989611198962
(ℎ)
(28)
where 119875(11989511198952)
1198941119894211989611198962
(119898ℎ)means that the probability of the 120574(119905)willnot reach to (119895
1 1198952) at times ℎ 2ℎ (119898 minus 1)ℎ but will reach
to (1198961 1198962) at time 119898ℎ Note that if ℎ le 119905 le 120575 then
120576 gt 1 minus 1198751198941119894211989411198942
(119905)
= sum
(1198961 1198962) = (1198941 1198942)
1198751198941119894211989611198962
(119905) ge 1198751198941119894211989511198952
(119905)
ge
119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ) 1198751198951119895211989511198952
(119905 minus 119898ℎ)
ge (1 minus 120576)
119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ)
(29)
which indicates119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ) le120576
1 minus 120576 (30)
Then making use of
1198751198941119894211989411198942
(119898ℎ) = 119875(11989511198952)
1198941119894211989411198942
(119898ℎ)
+
119898minus1
sum
119897=1
119875(11989511198952)
1198941119894211989511198952
(119897ℎ) 1198751198951119895211989411198942
((119898 minus 119897) ℎ)
(31)
we obtain
119875(11989511198952)
1198941119894211989411198942
(119898ℎ) ge 1198751198941119894211989411198942
(119898ℎ) minus
119898minus1
sum
119897=1
119875(11989511198952)
1198941119894211989511198952
(119897ℎ)
ge 1 minus 120576 minus120576
1 minus 120576
(32)
Consequently
1198751198941119894211989511198952
(119905)
gt
119899
sum
119898=1
119875(11989511198952)
1198941119894211989411198942
((119898 minus 1) ℎ) 1198751198941119894211989511198952
(ℎ) 1198751198951119895211989511198952
(119905 minus 119898ℎ)
ge 119899 (1 minus 120576 minus120576
1 minus 120576)1198751198941119894211989511198952
(ℎ) (1 minus 120576)
ge 119899 (1 minus 3120576) 1198751198941119894211989511198952
(ℎ)
(33)
Dividing both sides of the above inequality by ℎ and noting119899ℎ rarr 119905 whenever ℎ rarr 0 yield
lim supℎrarr0
1198751198941119894211989511198952
(ℎ)
ℎle
1
1 minus 3120576
1198751198941119894211989511198952
(119905)
119905lt infin (34)
Then letting 119905 rarr 0 gives
lim supℎrarr0
1198751198941119894211989511198952
(ℎ)
ℎle
1
1 minus 3120576lim inf119905rarr0
1198751198941119894211989511198952
(119905)
119905 (35)
and the required assertion follows immediately by letting120576 rarr 0 This completes the proof
Set 120574(119905) = (1205741(119905) 1205742(119905)) then it is easy to see that
almost every sample path of 120574(119905) is a right continuous stepfunction Now letting P(119905) = (119875
1198941119894211989511198952
(119905))11987311198732times11987311198732
Q =
(1199021198941119894211989511198952
)11987311198732times11987311198732
= P1015840(0) Then by Chapman-Kolmogorovequation
P (119905 + ℎ) = P (119905)P (ℎ) = P (ℎ)P (119905) (36)
we have
P (119905 + ℎ) minus P (119905)
ℎ= P (119905) [
P (ℎ) minus Iℎ
] = [P (ℎ) minus I
ℎ]P (119905)
(37)
Letting ℎ rarr 0 and taking limits give
P1015840(119905) = P (119905)Q
P1015840(119905) = QP (119905)
(38)
Note that
P (0) = I (39)
Then by solving the ordinary differential equations (38) and(39) we obtain the following lemma
Lemma 5 For P(119905) and Q one has
P (119905) = exp Q119905 (40)
We are now in the position to give the sufficient condi-tions for the existence and uniqueness of the solution of (8)For this end let us first give the definition of the solution
Journal of Applied Mathematics 5
Definition 6 An R119899-valued stochastic process 119909(119905)1199050le119905le119879
iscalled a solution of (8) if it has the following properties
(i) 119909(119905)1199050le119905le119879
is continuous andF119905-adapted
(ii) 119891(119909(119905) 119905 1205741(119905))1199050le119905le119879
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(53)
Consequently
1 + 119864( sup1199050le119905le120591119896
|119909 (119905)|2)
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(54)
Then the required inequality (44) follows immediately byletting 119896 rarr infin
Condition (42) indicates that the coefficients 119891(119909 119905 1198941)
and 119892(119909 119905 1198942) do not change faster than a linear function of
119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894
1) and 119892(119909 119905 119894
2) in 119909 for all 119905 isin [119905
0 119879] Then functions
that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution
Definition 8 Let120590infinbe a stopping time such that 119905
0le 120590infin
le 119879
as An R119899-valuedF119905-adapted continuous stochastic process
119909(119905)1199050le119905lt120590infin
is called a local solution of (8) if 119909(1199050) = 1199090and
moreover there is a nondecreasing sequence 120590119896119896ge1
the existence-and-uniqueness theorem holds on every finitesubinterval [119905
0 119879] of [119905
0infin) then (69) has a unique solution
119909(119905) on the entire interval [1199050infin) Such a solution is called
a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
where 119909119894is the size of 119894th species at time 119905 119903
1198940(119895) represents
the growth rate of 119894th species in regime 119895 for 119894 = 1 2119895 isin S and 119861
1and 119861
2are independent standard Brownian
motions However there are many stochastic factors thataffect some coefficients intensely but have little impact onother coefficients in (6) For example suppose that thestochastic factor is rain falls and 119909
1is able to endure a damp
weather while 1199092is fond of a dry environment then the rain
falls will affect 1199092intensely but have little impact on 119909
Thus the above examples show that the study of thefollowing SDEs with multi-Markovian switchings is essentialand is of great importance fromboth theoretical and practicalpoints
switching among each other according to the movement ofthe Markovian chains It is important for us to discover the
properties of the system (8) and to find out whether thepresence of two Markovian switchings affects some knownresults The first step and the foundation of those studies areto establish the theorems for the existence and uniqueness ofthe solution to system (8) So in this paper we will give sometheorems for the existence and uniqueness of the solution tosystem (8) and study some properties of this solution Thetheory developed in this paper is the foundation for furtherstudy and can be applied in many different and complicatedsituations and hence the importance of the results in thispaper is clear
It should be pointed out that the theory developed in thispaper can be generalized to cope with the more general SDEswith more Markovian chains
The reason we concentrate on (8) rather than (11) is to avoidthe notations becoming too complicated Once the theorydeveloped in this paper is established the reader should beable to cope with the more general (11) without any difficulty
The remaining part of this paper is as follows In Section 2the sufficient criteria for existence anduniqueness of solutionlocal solution andmaximal local solution will be establishedrespectively In Section 3 the 119871
119875-estimates of the solutionwill be given In Section 4 we will introduce an example toillustrate our main result Finally we will close the paper withconclusions in Section 5
2 SDEs with Markovian Chains
Throughout this paper let (ΩF F119905119905isin119877+
P) be a completeprobability space Let 119861(119905) = (119861
1(119905) 119861
119898(119905))119879 be an 119898-
dimensional Brownian motion defined on the probabilityspace
In this section we will consider (8) Let 120574(119905) = (1205741(119905)
1205742(119905)) We impose a hypothesis(H1) 120574
1(119905) is independent of 120574
2(119905)
Then 120574(119905) is a homogenous vector Markovian chain withtransition probabilities
This implies minus1199021198941119894211989411198942
le lim inf119905rarr0
(120601(119905)119905) Thus
minus1199021198941119894211989411198942
= lim119905rarr0
120601 (119905)
119905 (24)
Using the definition of 120601(119905) gives
lim119905rarr0
1 minus 1198751198941119894211989411198942
(119905)
119905= lim119905rarr0
1 minus exp 120601 (119905)
120601 (119905)
120601 (119905)
119905= minus1199021198941119894211989411198942
(25)
which is the required assertion
4 Journal of Applied Mathematics
Lemma 4 Under Assumption (H2) for (1198941 1198942) (1198951 1198952) isin
S (1198941 1198942) = (1198951 1198952)
1199021198941119894211989511198952
= 1198751015840
1198941119894211989511198952
(0) = lim119905rarr0
1198751198941119894211989511198952
(119905)
119905(26)
exists and is finite
Proof By (H2) we note that for all 0 lt 120576 lt 13 exist0 lt 120575 lt 1such that
1198751198941119894211989411198942
(119905) gt 1 minus 120576 1198751198951119895211989511198952
(119905) gt 1 minus 120576
1198751198951119895211989411198942
(119905) lt 120576
(27)
provided 0 lt 119905 le 120575For forall0 le ℎ lt 119905 set 119899 = ⟨119905ℎ⟩ where ⟨119886⟩ = max
119899le119886119899 isin
Z Let
119875(11989511198952)
1198941119894211989611198962
(ℎ) = 1198751198941119894211989611198962
(ℎ)
119875(11989511198952)
1198941119894211989611198962
(119898ℎ)
= sum
(11990311199032) = (1198951 1198952)
119875(11989511198952)
1198941119894211990311199032
((119898 minus 1) ℎ) 1198751199031119903211989611198962
(ℎ)
(28)
where 119875(11989511198952)
1198941119894211989611198962
(119898ℎ)means that the probability of the 120574(119905)willnot reach to (119895
1 1198952) at times ℎ 2ℎ (119898 minus 1)ℎ but will reach
to (1198961 1198962) at time 119898ℎ Note that if ℎ le 119905 le 120575 then
120576 gt 1 minus 1198751198941119894211989411198942
(119905)
= sum
(1198961 1198962) = (1198941 1198942)
1198751198941119894211989611198962
(119905) ge 1198751198941119894211989511198952
(119905)
ge
119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ) 1198751198951119895211989511198952
(119905 minus 119898ℎ)
ge (1 minus 120576)
119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ)
(29)
which indicates119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ) le120576
1 minus 120576 (30)
Then making use of
1198751198941119894211989411198942
(119898ℎ) = 119875(11989511198952)
1198941119894211989411198942
(119898ℎ)
+
119898minus1
sum
119897=1
119875(11989511198952)
1198941119894211989511198952
(119897ℎ) 1198751198951119895211989411198942
((119898 minus 119897) ℎ)
(31)
we obtain
119875(11989511198952)
1198941119894211989411198942
(119898ℎ) ge 1198751198941119894211989411198942
(119898ℎ) minus
119898minus1
sum
119897=1
119875(11989511198952)
1198941119894211989511198952
(119897ℎ)
ge 1 minus 120576 minus120576
1 minus 120576
(32)
Consequently
1198751198941119894211989511198952
(119905)
gt
119899
sum
119898=1
119875(11989511198952)
1198941119894211989411198942
((119898 minus 1) ℎ) 1198751198941119894211989511198952
(ℎ) 1198751198951119895211989511198952
(119905 minus 119898ℎ)
ge 119899 (1 minus 120576 minus120576
1 minus 120576)1198751198941119894211989511198952
(ℎ) (1 minus 120576)
ge 119899 (1 minus 3120576) 1198751198941119894211989511198952
(ℎ)
(33)
Dividing both sides of the above inequality by ℎ and noting119899ℎ rarr 119905 whenever ℎ rarr 0 yield
lim supℎrarr0
1198751198941119894211989511198952
(ℎ)
ℎle
1
1 minus 3120576
1198751198941119894211989511198952
(119905)
119905lt infin (34)
Then letting 119905 rarr 0 gives
lim supℎrarr0
1198751198941119894211989511198952
(ℎ)
ℎle
1
1 minus 3120576lim inf119905rarr0
1198751198941119894211989511198952
(119905)
119905 (35)
and the required assertion follows immediately by letting120576 rarr 0 This completes the proof
Set 120574(119905) = (1205741(119905) 1205742(119905)) then it is easy to see that
almost every sample path of 120574(119905) is a right continuous stepfunction Now letting P(119905) = (119875
1198941119894211989511198952
(119905))11987311198732times11987311198732
Q =
(1199021198941119894211989511198952
)11987311198732times11987311198732
= P1015840(0) Then by Chapman-Kolmogorovequation
P (119905 + ℎ) = P (119905)P (ℎ) = P (ℎ)P (119905) (36)
we have
P (119905 + ℎ) minus P (119905)
ℎ= P (119905) [
P (ℎ) minus Iℎ
] = [P (ℎ) minus I
ℎ]P (119905)
(37)
Letting ℎ rarr 0 and taking limits give
P1015840(119905) = P (119905)Q
P1015840(119905) = QP (119905)
(38)
Note that
P (0) = I (39)
Then by solving the ordinary differential equations (38) and(39) we obtain the following lemma
Lemma 5 For P(119905) and Q one has
P (119905) = exp Q119905 (40)
We are now in the position to give the sufficient condi-tions for the existence and uniqueness of the solution of (8)For this end let us first give the definition of the solution
Journal of Applied Mathematics 5
Definition 6 An R119899-valued stochastic process 119909(119905)1199050le119905le119879
iscalled a solution of (8) if it has the following properties
(i) 119909(119905)1199050le119905le119879
is continuous andF119905-adapted
(ii) 119891(119909(119905) 119905 1205741(119905))1199050le119905le119879
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(53)
Consequently
1 + 119864( sup1199050le119905le120591119896
|119909 (119905)|2)
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(54)
Then the required inequality (44) follows immediately byletting 119896 rarr infin
Condition (42) indicates that the coefficients 119891(119909 119905 1198941)
and 119892(119909 119905 1198942) do not change faster than a linear function of
119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894
1) and 119892(119909 119905 119894
2) in 119909 for all 119905 isin [119905
0 119879] Then functions
that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution
Definition 8 Let120590infinbe a stopping time such that 119905
0le 120590infin
le 119879
as An R119899-valuedF119905-adapted continuous stochastic process
119909(119905)1199050le119905lt120590infin
is called a local solution of (8) if 119909(1199050) = 1199090and
moreover there is a nondecreasing sequence 120590119896119896ge1
the existence-and-uniqueness theorem holds on every finitesubinterval [119905
0 119879] of [119905
0infin) then (69) has a unique solution
119909(119905) on the entire interval [1199050infin) Such a solution is called
a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
This implies minus1199021198941119894211989411198942
le lim inf119905rarr0
(120601(119905)119905) Thus
minus1199021198941119894211989411198942
= lim119905rarr0
120601 (119905)
119905 (24)
Using the definition of 120601(119905) gives
lim119905rarr0
1 minus 1198751198941119894211989411198942
(119905)
119905= lim119905rarr0
1 minus exp 120601 (119905)
120601 (119905)
120601 (119905)
119905= minus1199021198941119894211989411198942
(25)
which is the required assertion
4 Journal of Applied Mathematics
Lemma 4 Under Assumption (H2) for (1198941 1198942) (1198951 1198952) isin
S (1198941 1198942) = (1198951 1198952)
1199021198941119894211989511198952
= 1198751015840
1198941119894211989511198952
(0) = lim119905rarr0
1198751198941119894211989511198952
(119905)
119905(26)
exists and is finite
Proof By (H2) we note that for all 0 lt 120576 lt 13 exist0 lt 120575 lt 1such that
1198751198941119894211989411198942
(119905) gt 1 minus 120576 1198751198951119895211989511198952
(119905) gt 1 minus 120576
1198751198951119895211989411198942
(119905) lt 120576
(27)
provided 0 lt 119905 le 120575For forall0 le ℎ lt 119905 set 119899 = ⟨119905ℎ⟩ where ⟨119886⟩ = max
119899le119886119899 isin
Z Let
119875(11989511198952)
1198941119894211989611198962
(ℎ) = 1198751198941119894211989611198962
(ℎ)
119875(11989511198952)
1198941119894211989611198962
(119898ℎ)
= sum
(11990311199032) = (1198951 1198952)
119875(11989511198952)
1198941119894211990311199032
((119898 minus 1) ℎ) 1198751199031119903211989611198962
(ℎ)
(28)
where 119875(11989511198952)
1198941119894211989611198962
(119898ℎ)means that the probability of the 120574(119905)willnot reach to (119895
1 1198952) at times ℎ 2ℎ (119898 minus 1)ℎ but will reach
to (1198961 1198962) at time 119898ℎ Note that if ℎ le 119905 le 120575 then
120576 gt 1 minus 1198751198941119894211989411198942
(119905)
= sum
(1198961 1198962) = (1198941 1198942)
1198751198941119894211989611198962
(119905) ge 1198751198941119894211989511198952
(119905)
ge
119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ) 1198751198951119895211989511198952
(119905 minus 119898ℎ)
ge (1 minus 120576)
119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ)
(29)
which indicates119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ) le120576
1 minus 120576 (30)
Then making use of
1198751198941119894211989411198942
(119898ℎ) = 119875(11989511198952)
1198941119894211989411198942
(119898ℎ)
+
119898minus1
sum
119897=1
119875(11989511198952)
1198941119894211989511198952
(119897ℎ) 1198751198951119895211989411198942
((119898 minus 119897) ℎ)
(31)
we obtain
119875(11989511198952)
1198941119894211989411198942
(119898ℎ) ge 1198751198941119894211989411198942
(119898ℎ) minus
119898minus1
sum
119897=1
119875(11989511198952)
1198941119894211989511198952
(119897ℎ)
ge 1 minus 120576 minus120576
1 minus 120576
(32)
Consequently
1198751198941119894211989511198952
(119905)
gt
119899
sum
119898=1
119875(11989511198952)
1198941119894211989411198942
((119898 minus 1) ℎ) 1198751198941119894211989511198952
(ℎ) 1198751198951119895211989511198952
(119905 minus 119898ℎ)
ge 119899 (1 minus 120576 minus120576
1 minus 120576)1198751198941119894211989511198952
(ℎ) (1 minus 120576)
ge 119899 (1 minus 3120576) 1198751198941119894211989511198952
(ℎ)
(33)
Dividing both sides of the above inequality by ℎ and noting119899ℎ rarr 119905 whenever ℎ rarr 0 yield
lim supℎrarr0
1198751198941119894211989511198952
(ℎ)
ℎle
1
1 minus 3120576
1198751198941119894211989511198952
(119905)
119905lt infin (34)
Then letting 119905 rarr 0 gives
lim supℎrarr0
1198751198941119894211989511198952
(ℎ)
ℎle
1
1 minus 3120576lim inf119905rarr0
1198751198941119894211989511198952
(119905)
119905 (35)
and the required assertion follows immediately by letting120576 rarr 0 This completes the proof
Set 120574(119905) = (1205741(119905) 1205742(119905)) then it is easy to see that
almost every sample path of 120574(119905) is a right continuous stepfunction Now letting P(119905) = (119875
1198941119894211989511198952
(119905))11987311198732times11987311198732
Q =
(1199021198941119894211989511198952
)11987311198732times11987311198732
= P1015840(0) Then by Chapman-Kolmogorovequation
P (119905 + ℎ) = P (119905)P (ℎ) = P (ℎ)P (119905) (36)
we have
P (119905 + ℎ) minus P (119905)
ℎ= P (119905) [
P (ℎ) minus Iℎ
] = [P (ℎ) minus I
ℎ]P (119905)
(37)
Letting ℎ rarr 0 and taking limits give
P1015840(119905) = P (119905)Q
P1015840(119905) = QP (119905)
(38)
Note that
P (0) = I (39)
Then by solving the ordinary differential equations (38) and(39) we obtain the following lemma
Lemma 5 For P(119905) and Q one has
P (119905) = exp Q119905 (40)
We are now in the position to give the sufficient condi-tions for the existence and uniqueness of the solution of (8)For this end let us first give the definition of the solution
Journal of Applied Mathematics 5
Definition 6 An R119899-valued stochastic process 119909(119905)1199050le119905le119879
iscalled a solution of (8) if it has the following properties
(i) 119909(119905)1199050le119905le119879
is continuous andF119905-adapted
(ii) 119891(119909(119905) 119905 1205741(119905))1199050le119905le119879
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(53)
Consequently
1 + 119864( sup1199050le119905le120591119896
|119909 (119905)|2)
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(54)
Then the required inequality (44) follows immediately byletting 119896 rarr infin
Condition (42) indicates that the coefficients 119891(119909 119905 1198941)
and 119892(119909 119905 1198942) do not change faster than a linear function of
119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894
1) and 119892(119909 119905 119894
2) in 119909 for all 119905 isin [119905
0 119879] Then functions
that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution
Definition 8 Let120590infinbe a stopping time such that 119905
0le 120590infin
le 119879
as An R119899-valuedF119905-adapted continuous stochastic process
119909(119905)1199050le119905lt120590infin
is called a local solution of (8) if 119909(1199050) = 1199090and
moreover there is a nondecreasing sequence 120590119896119896ge1
the existence-and-uniqueness theorem holds on every finitesubinterval [119905
0 119879] of [119905
0infin) then (69) has a unique solution
119909(119905) on the entire interval [1199050infin) Such a solution is called
a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
Lemma 4 Under Assumption (H2) for (1198941 1198942) (1198951 1198952) isin
S (1198941 1198942) = (1198951 1198952)
1199021198941119894211989511198952
= 1198751015840
1198941119894211989511198952
(0) = lim119905rarr0
1198751198941119894211989511198952
(119905)
119905(26)
exists and is finite
Proof By (H2) we note that for all 0 lt 120576 lt 13 exist0 lt 120575 lt 1such that
1198751198941119894211989411198942
(119905) gt 1 minus 120576 1198751198951119895211989511198952
(119905) gt 1 minus 120576
1198751198951119895211989411198942
(119905) lt 120576
(27)
provided 0 lt 119905 le 120575For forall0 le ℎ lt 119905 set 119899 = ⟨119905ℎ⟩ where ⟨119886⟩ = max
119899le119886119899 isin
Z Let
119875(11989511198952)
1198941119894211989611198962
(ℎ) = 1198751198941119894211989611198962
(ℎ)
119875(11989511198952)
1198941119894211989611198962
(119898ℎ)
= sum
(11990311199032) = (1198951 1198952)
119875(11989511198952)
1198941119894211990311199032
((119898 minus 1) ℎ) 1198751199031119903211989611198962
(ℎ)
(28)
where 119875(11989511198952)
1198941119894211989611198962
(119898ℎ)means that the probability of the 120574(119905)willnot reach to (119895
1 1198952) at times ℎ 2ℎ (119898 minus 1)ℎ but will reach
to (1198961 1198962) at time 119898ℎ Note that if ℎ le 119905 le 120575 then
120576 gt 1 minus 1198751198941119894211989411198942
(119905)
= sum
(1198961 1198962) = (1198941 1198942)
1198751198941119894211989611198962
(119905) ge 1198751198941119894211989511198952
(119905)
ge
119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ) 1198751198951119895211989511198952
(119905 minus 119898ℎ)
ge (1 minus 120576)
119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ)
(29)
which indicates119899
sum
119898=1
119875(11989511198952)
1198941119894211989511198952
(119898ℎ) le120576
1 minus 120576 (30)
Then making use of
1198751198941119894211989411198942
(119898ℎ) = 119875(11989511198952)
1198941119894211989411198942
(119898ℎ)
+
119898minus1
sum
119897=1
119875(11989511198952)
1198941119894211989511198952
(119897ℎ) 1198751198951119895211989411198942
((119898 minus 119897) ℎ)
(31)
we obtain
119875(11989511198952)
1198941119894211989411198942
(119898ℎ) ge 1198751198941119894211989411198942
(119898ℎ) minus
119898minus1
sum
119897=1
119875(11989511198952)
1198941119894211989511198952
(119897ℎ)
ge 1 minus 120576 minus120576
1 minus 120576
(32)
Consequently
1198751198941119894211989511198952
(119905)
gt
119899
sum
119898=1
119875(11989511198952)
1198941119894211989411198942
((119898 minus 1) ℎ) 1198751198941119894211989511198952
(ℎ) 1198751198951119895211989511198952
(119905 minus 119898ℎ)
ge 119899 (1 minus 120576 minus120576
1 minus 120576)1198751198941119894211989511198952
(ℎ) (1 minus 120576)
ge 119899 (1 minus 3120576) 1198751198941119894211989511198952
(ℎ)
(33)
Dividing both sides of the above inequality by ℎ and noting119899ℎ rarr 119905 whenever ℎ rarr 0 yield
lim supℎrarr0
1198751198941119894211989511198952
(ℎ)
ℎle
1
1 minus 3120576
1198751198941119894211989511198952
(119905)
119905lt infin (34)
Then letting 119905 rarr 0 gives
lim supℎrarr0
1198751198941119894211989511198952
(ℎ)
ℎle
1
1 minus 3120576lim inf119905rarr0
1198751198941119894211989511198952
(119905)
119905 (35)
and the required assertion follows immediately by letting120576 rarr 0 This completes the proof
Set 120574(119905) = (1205741(119905) 1205742(119905)) then it is easy to see that
almost every sample path of 120574(119905) is a right continuous stepfunction Now letting P(119905) = (119875
1198941119894211989511198952
(119905))11987311198732times11987311198732
Q =
(1199021198941119894211989511198952
)11987311198732times11987311198732
= P1015840(0) Then by Chapman-Kolmogorovequation
P (119905 + ℎ) = P (119905)P (ℎ) = P (ℎ)P (119905) (36)
we have
P (119905 + ℎ) minus P (119905)
ℎ= P (119905) [
P (ℎ) minus Iℎ
] = [P (ℎ) minus I
ℎ]P (119905)
(37)
Letting ℎ rarr 0 and taking limits give
P1015840(119905) = P (119905)Q
P1015840(119905) = QP (119905)
(38)
Note that
P (0) = I (39)
Then by solving the ordinary differential equations (38) and(39) we obtain the following lemma
Lemma 5 For P(119905) and Q one has
P (119905) = exp Q119905 (40)
We are now in the position to give the sufficient condi-tions for the existence and uniqueness of the solution of (8)For this end let us first give the definition of the solution
Journal of Applied Mathematics 5
Definition 6 An R119899-valued stochastic process 119909(119905)1199050le119905le119879
iscalled a solution of (8) if it has the following properties
(i) 119909(119905)1199050le119905le119879
is continuous andF119905-adapted
(ii) 119891(119909(119905) 119905 1205741(119905))1199050le119905le119879
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(53)
Consequently
1 + 119864( sup1199050le119905le120591119896
|119909 (119905)|2)
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(54)
Then the required inequality (44) follows immediately byletting 119896 rarr infin
Condition (42) indicates that the coefficients 119891(119909 119905 1198941)
and 119892(119909 119905 1198942) do not change faster than a linear function of
119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894
1) and 119892(119909 119905 119894
2) in 119909 for all 119905 isin [119905
0 119879] Then functions
that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution
Definition 8 Let120590infinbe a stopping time such that 119905
0le 120590infin
le 119879
as An R119899-valuedF119905-adapted continuous stochastic process
119909(119905)1199050le119905lt120590infin
is called a local solution of (8) if 119909(1199050) = 1199090and
moreover there is a nondecreasing sequence 120590119896119896ge1
the existence-and-uniqueness theorem holds on every finitesubinterval [119905
0 119879] of [119905
0infin) then (69) has a unique solution
119909(119905) on the entire interval [1199050infin) Such a solution is called
a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(53)
Consequently
1 + 119864( sup1199050le119905le120591119896
|119909 (119905)|2)
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(54)
Then the required inequality (44) follows immediately byletting 119896 rarr infin
Condition (42) indicates that the coefficients 119891(119909 119905 1198941)
and 119892(119909 119905 1198942) do not change faster than a linear function of
119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894
1) and 119892(119909 119905 119894
2) in 119909 for all 119905 isin [119905
0 119879] Then functions
that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution
Definition 8 Let120590infinbe a stopping time such that 119905
0le 120590infin
le 119879
as An R119899-valuedF119905-adapted continuous stochastic process
119909(119905)1199050le119905lt120590infin
is called a local solution of (8) if 119909(1199050) = 1199090and
moreover there is a nondecreasing sequence 120590119896119896ge1
the existence-and-uniqueness theorem holds on every finitesubinterval [119905
0 119879] of [119905
0infin) then (69) has a unique solution
119909(119905) on the entire interval [1199050infin) Such a solution is called
a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(53)
Consequently
1 + 119864( sup1199050le119905le120591119896
|119909 (119905)|2)
le (1 + 311986410038161003816100381610038161199090
10038161003816100381610038162) exp 3119870 (119879 minus 119905
0) (119879 minus 119905
0+ 4)
(54)
Then the required inequality (44) follows immediately byletting 119896 rarr infin
Condition (42) indicates that the coefficients 119891(119909 119905 1198941)
and 119892(119909 119905 1198942) do not change faster than a linear function of
119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894
1) and 119892(119909 119905 119894
2) in 119909 for all 119905 isin [119905
0 119879] Then functions
that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution
Definition 8 Let120590infinbe a stopping time such that 119905
0le 120590infin
le 119879
as An R119899-valuedF119905-adapted continuous stochastic process
119909(119905)1199050le119905lt120590infin
is called a local solution of (8) if 119909(1199050) = 1199090and
moreover there is a nondecreasing sequence 120590119896119896ge1
the existence-and-uniqueness theorem holds on every finitesubinterval [119905
0 119879] of [119905
0infin) then (69) has a unique solution
119909(119905) on the entire interval [1199050infin) Such a solution is called
a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
the existence-and-uniqueness theorem holds on every finitesubinterval [119905
0 119879] of [119905
0infin) then (69) has a unique solution
119909(119905) on the entire interval [1199050infin) Such a solution is called
a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
Then the required assertion follows from the Gronwallinequality
Up to now we have discussed the 119871119901-estimates for the
solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies
where 1205741(119905) is a right-continuous homogenous Markovian
chain taking values in finite state spaces S1= 1 2 and 120574
2(119905)
is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S
2= 1 2 3 120583(119894) = 119894 119894 = 1 2
](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution
5 Conclusions and Further Research
This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis
(H1)1015840 both 1205741(119905) and 120574
2(119905) are right-continuous homoge-
nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a
homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper
can be established similarly It is easy to see that if 1205741(119905) equiv
1205742(119905) and 120574
1(119905) is a right-continuous homogenous Markovian
chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574
2(119905) (8) will reduce to the classical SDEs with
single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions
Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress
Acknowledgments
The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)
References
[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989
[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006
[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004
[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002
[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009
[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007
[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009
[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal
Journal of Applied Mathematics 11
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012
ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011
[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011
[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012
[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012