Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2. Combined Approach Based on Generalized Multiple and Iterated Fourier Series

TL;DR

This paper develops a combined Fourier series approach to expand iterated Stratonovich stochastic integrals of multiplicity 2, proving mean-square convergence and enabling improved numerical solutions for stochastic differential equations.

Contribution

It introduces a novel combined Fourier series method for expanding iterated Stratonovich integrals and proves their convergence properties, advancing stochastic calculus techniques.

Findings

Proved mean-square convergence of the first expansion part.

Established pointwise convergence of the second expansion part.

Applied results to numerical integration of Ito stochastic differential equations.

Abstract

The article is devoted to the expansion of iterated Stratonovich stochastic integrals of multiplicity 2 on the base of the combined approach of generalized multiple and iterated Fourier series. We consider two different parts of the expansion of iterated Stratonovich stochastic integrals. The mean-square convergence of the first part is proved on the base of generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^2).$ The mean-square convergence of the second part is proved on the base of generalized iterated (double) Fourier series converging pointwise. At that, we prove the iterated limit transition for the second part of the expansion on the base of Lebesgue's Dominated Convergence Theorem. The results of the article can be applied to the numerical integration of Ito stochastic differential equations.