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Approximation of stochastic differential equations driven by α-stable Lévy motion

100%

Janicki A. , Michna Z. , Weron A.

Applicationes Mathematicae

1996-1997

tom 24

nr 2

149-168

In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to α-stable Lévy motion. We prove an appropriate weak limit theorem, which does not follow from known results on stability properties of stochastic differential equations driven by semimartingales. It assures convergence in law in the Skorokhod topology of sequences of approximate solutions and justifies discrete time schemes applied in computer simulations. An example is included in order to demonstrate that stochastic differential equations with jumps are of interest in constructions of models for various problems arising in science and engineering, often providing better description of real life phenomena than their Gaussian counterparts. In order to demonstrate the usefulness of our approach, we present computer simulations of a continuous time α-stable model of cumulative gain in the Duffie-Harrison option pricing framework.

Small deviation of subordinated processes over compact sets

88%

Linde W. , Zipfel P.

Probability and Mathematical Statistics

2008

tom Vol. 28, Fasc. 2

281--304

Let A =(A(t)t≥0 be a subordinator. Given a compact set K ⊂[(0;∞) we prove two-sided estimates for the covering numbers of the random set {A(t) : t ∈ K} which depend on the Laplace exponent Φ of A and on the covering numbers of K. This extends former results in the case K = [0; 1]. Using this we find the behavior of the small deviation probabilities for subordinated processes(WH(A(t))tЄK, whereWH is a fractional Brownian motion with Hurst index 0 < H < 1. The results are valid in the quenched as well as in the annealed case. In particular, those questions are investigated for Gamma processes. Here some surprising new phenomena appear. As application of the general results we find the behavior of log P(suptЄK |Zα(t)| < ε) as ε→ 0 for the α-stable Lévy motion Zα. For example, if K is a self-similar set with Hausdorff dimension D > 0, then this behavior is of order −ε−αD in complete accordance with the Gaussian case α = 2.

Approximation of finite-dimensional distributions for integrals driven by α-stable Lévy motion

88%

Janicki A.

Applicationes Mathematicae

1998-1999

tom 25

nr 4

473-488

We present a method of numerical approximation for stochastic integrals involving α-stable Lévy motion as an integrator. Constructions of approximate sums are based on the Poissonian series representation of such random measures. The main result gives an estimate of the rate of convergence of finite-dimensional distributions of finite sums approximating such stochastic integrals. Stochastic integrals driven by such measures are of interest in constructions of models for various problems arising in science and engineering, often providing a better description of real life phenomena than their Gaussian counterparts.