Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals No` elia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico Scalas BCAM Seminar - 26 July 2012 No` elia Viles (BCAM) BCAM seminar 26 July 2012 1 / 43
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Functional Limit theorems for the quadratic variation ofa continuous time random walk and for certain
stochastic integrals
Noelia Viles CuadrosBCAM- Basque Center of Applied Mathematics
The Skorokhod space provides a natural and convenient formalism fordescribing the trajectories of stochastic processes with jumps: Poissonprocess, Levy processes, martingales and semimartingales, empiricaldistribution functions, discretizations of stochastic processes, etc.
It can be assigned a topology that, intuitively allows us to wiggle spaceand time a bit (whereas the traditional topology of uniform convergenceonly allows us to wiggle space a bit).
Skorokhod (1965) proposed four metric separable topologies on D,denoted by J1, J2, M1 and M2.
A. Skorokhod.Limit Theorems for Stochastic Processes.Theor. Probability Appl. 1, 261–290, 1956.
The sequence xn(t) ∈ D converges to x0(t) ∈ D in the M1-topology if
limn→+∞
dM1(xn(t), x0(t)) = 0. (4)
In other words, we have the convergence in M1-topology if there existparametric representations (y(s), t(s)) of the graph Γx0(t) and(yn(s), tn(s)) of the graph Γxn(t) such that
A continuous time random walk (CTRW) is a pure jump process given bya sum of i.i.d. random jumps (Yi )i∈N separated by i.i.d. random waitingtimes (positive random variables) (Ji )i∈N.
A continuous-time process L = Ltt>0 with values in R is called a Levyprocess if its sample paths are cadlag at every time point t, and it hasstationary, independent increments, that is:
(a) For all 0 = t0 < t1 < · · · < tk , the increments Lti − Lti−1 areindependent.
(b) For all 0 6 s 6 t the random variables Lt − Ls and Lt−s − L0 have thesame distribution.
An α-stable process is a real-valued Levy process Lα = Lα(t)t>0 withinitial value Lα(0) that satisfies the self-similarity property
1
t1/αLα(t)
L= Lα(1), ∀t > 0.
If α = 2 then the α-stable Levy process is the Wiener process.
Compound Fractional Poisson ProcessThe semi-Markov extension of the compound Poisson process is thecompound fractional Poisson process.
The counting process associated is called the fractional Poisson process
Nβ(t) = maxn : Tn 6 t.
For β ∈ (0, 1), the FPP is semi-Markov and it is not Levy .If we subordinate a CTRW to the fractional Poisson process, we obtain thecompound fractional Poisson process, which is not Markov
XNβ(t) =
Nβ(t)∑i=1
Yi . (11)
The functional limit of the compound fractional Poisson process is anα-stable Levy process subordinated to the fractional Poisson process.
These processes are possible models for tick-by-tick financial data.
M. Meerschaert, H. P. Scheffler.Limit Theorems for continuous time random walks.Available athttp://www.mathematik.uni-dortmund.de/lsiv/scheffler/ctrw1.pdf.,2001.
M. Meerschaert, H. P. Scheffler.Limit Distributions for Sums of Independent Random Vectors: HeavyTails in Theory and Practice.Wiley Series in Probability and Statistics., 2001.
Convergence to the symmetric α-stable Levy processAssume the jumps Yi belong to the strict generalized domain of attractionof some stable law with α ∈ (0, 2), then ∃an > 0 such that
Under the distributional assumptions considered above for the waitingtimes Ji and the jumps Yi , we havec−β/α
Nβ(t)∑i=1
Yi
t>0
M1−top⇒ Lα(D−1β (t))t>0, when c → +∞, (12)
in the Skorokhod space D([0,+∞),R) endowed with the M1-topology.
M. Meerschaert, H. P. Scheffler.Limit theorems for continuous-time random walks with infinite meanwaiting times.J. Appl. Probab., 41 (3), 623–638, 2004.
Suppose that (xn, yn)→ (x , y) in D([0, a],Rk)× D1↑ (where D1
↑ is the
subset of functions nondecreasing and with x i (0) > 0). If y is continuousand strictly increasing at t whenever y(t) ∈ Disc(x) and x is monotone on[y(t−), y(t)] and y(t−), y(t) /∈ Disc(x) whenever t ∈ Disc(y), thenxn yn → x y in D([0, a],Rk), where the topology throughout is M1 orM2.
In that case we can only ensure the convergence in the weaker topology,M1 instead of J1 because D−1
β (t)t>0 is not strictly increasing.
W. Whitt,Stochastic-Process Limits: An Introduction to Stochastic-ProcessLimits and Their Application to Queues.Springer, New York (2002).
E. Scalas, N. Viles,On the Convergence of Quadratic variation for Compound FractionalPoisson Processes.Fractional Calculus and Applied Analysis, 15, 314–331 (2012).
Damped harmonic oscillator subject to a random force
The equation of motion is informally given by
x(t) + γx(t) + kx(t) = ξ(t), (16)
where x(t) is the position of the oscillating particle with unit mass at timet, γ > 0 is the damping coefficient, k > 0 is the spring constant and ξ(t)represents white Levy noise (formal derivative symmetric Lα(t)).
I. M. Sokolov,Harmonic oscillator under Levy noise: Unexpected properties in thephase space.Phys. Rev. E. Stat. Nonlin Soft Matter Phys 83, 041118 (2011).
Replace the white noise with a sequence of instantaneous shots of randomamplitude at random times and then with an appropriate functional limitof this process.
The sequence of instantaneous shots of random amplitude at randomtimes can be expressed in terms of the formal derivative of a compoundrenewal process given by
Let f ∈ Cb(R). Let Yii∈N be i.i.d. symmetric α-stable random variables.Assume that Y1 belongs DOA of Sα, with α ∈ (1, 2]. Let Jii∈N be i.i.d.such that J1 belongs to the strict DOA of Sβ withβ ∈ (0, 1). Consider
We have studied the convergence of a class of stochastic integralswith respect to the Compound Fractional Poisson Process.
Under proper scaling hypotheses, these integrals converge to theintegrals w.r.t a symmetric α-stable process subordinated to theinverse β-stable subordinator.