Approximation of Infinitely Divisible Random Variables with Application to the Simulation of Stochastic Processes

Magnus Wiktorsson

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
2001

ISBN 91-628-4640-X
LUTFMS--1014--2001

Abstract:

This thesis consists of four papers A, B, C and D. Paper A and B treats the simulation of stochastic differential equations (SDEs). The research presented therein was triggered by the fact that there were not any efficient implementations of the higher order methods for simulating SDEs. So in practice the higher order methods required at least the same amount of work as the Euler method to obtain a given mean square error. The faster convergence rate of the higher order methods requires the simulation of the so called iterated Itô integrals. In (A) we use a

shot-noise type series representation of one iterated Itô integral. We split the series representation into a sum of n terms and a remainder term and show that the remainder term is asymptotically Gaussian as n goes to infinity. We provide an explicit coupling of the remainder a Gaussian random variable and show that this improves the mean square error by a factor n^½. In (B) we provide a multi-dimensional extension of the results in (A) as well as the not previously known simultaneous characteristic function of all iterated Itô integrals obtained then pairing m independent Wiener processes. In (C) we study the simulation of type G Lévy processes. Recall that random variable is said to be of type G if it is a Gaussian variance mixture. We note that type G Lévy processes are subordinated Wiener processes. We use a series representation of the subordinator, a tail-sum approximation and obtain an explicit coupling between type G Lévy processes and the sum of a compound Poisson process and a scaled Wiener process. We calculate the mean integrated square error for this approximation. We examine the

convergence of the scaled tail-sum process to its mean value function and provide a sufficient condition for this convergence. In paper (D) we utilise the coupling results from paper (C) to obtain approximationsof stochastic integrals with respect to type G Lévy processes. Depending on the properties of the integrator we obtain either point-wise mean square error results or mean integrated square error results for the approximation. We also show that a stochastic time change representation of stochastic integrals can be used to obtain useful approximations.

Keywords:

Infinitely divisible distribution, stochastic differential equation, type G distribution, Lévy process, stochastic integral