Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series

TL;DR

This paper develops generalized Fourier series expansions for iterated stochastic integrals with respect to martingale Poisson measures and martingales, proving mean-square convergence and illustrating with Bessel functions.

Contribution

It introduces new expansions for iterated stochastic integrals with respect to martingales, extending previous Ito integral expansions using generalized multiple Fourier series.

Findings

Established expansions for iterated integrals with respect to martingale Poisson measures.

Proved mean-square convergence of the new expansions.

Provided an example using Bessel functions for double stochastic integrals.

Abstract

We consider some versions and generalizations of an approach to the expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series. Expansions of iterated stochastic integrals with respect to martingale Poisson measures and with respect to martingales were obtained. For the iterated stochastic integrals with respect to martingales we have proved theorem, which is a generalization of the expansion for iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series. Also we consider a modification of the mentioned expansion of iterated Ito stochastic integrals for the case of complete orthonormal with weight $r(t_1)\ldots r(t_k)\ge 0$ systems of functions in the space $L_2([t, T]^k)$. Mean-square convergence of the considered expansions is proved. An example of the expansion of iterated (double) stochastic integrals with respect to martingales using the system of Bessel functions is considered.