Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Iterated Fourier Series Converging Pointwise

TL;DR

This paper develops a method to expand iterated Stratonovich stochastic integrals of any multiplicity using generalized Fourier series, enabling their representation as series of Gaussian variables with proven convergence, aiding numerical solutions of stochastic differential equations.

Contribution

It introduces a pointwise convergent expansion of iterated Stratonovich integrals of arbitrary multiplicity using generalized Fourier series, including Fourier-Legendre and trigonometric series.

Findings

Expansion allows representation as series of Gaussian variables

Proves mean-square convergence of the series

Provides recent results for multiplicities 3 to 6

Abstract

The article is devoted to the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on the generalized iterated Fourier series converging pointwise. The case of Fourier-Legendre series as well as the case of trigonometric Fourier series are considered in details. The obtained expansion provides a possibility to represent the iterated Stratonovich stochastic integral in the form of iterated series of products of standard Gaussian random variables. Convergence in the mean of degree $2n$ $(n\in \mathbb{N})$ of the expansion is proved. Some recent results on the expansion of iterated Stratonovich stochastic integrals of multiplicities 3 to 6 are given. The results of the article can be applied to the numerical solution of Ito stochastic differential equations.