Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Multiple Fourier Series Converging in the Mean

TL;DR

This paper introduces a new method for expanding iterated Ito stochastic integrals of any multiplicity using generalized multiple Fourier series, ensuring mean-square convergence with only one limit operation, and generalizes to various orthonormal systems.

Contribution

The paper develops a novel expansion method for iterated Ito integrals based on generalized Fourier series with improved convergence properties and broader applicability to different orthonormal systems.

Findings

The method converges in mean-square with only one limit transition.

It generalizes to arbitrary orthonormal systems in $L_2$ spaces.

Proved convergence in mean of degree 2n and almost sure convergence.

Abstract

The article is devoted to the expansions of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the sense of norm in the space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ The method of generalized multiple Fourier series for expansion and mean-square approximation of iterated Ito stochastic integrals of arbitrary multiplicity $k$ ($k\in\mathbb{N}$) with respect to components of the multidimensional Wiener process is proposed and developed. The obtained expansions contain only one operation of the limit transition in contrast to its existing analogues. In the article it is also obtained the generalization of the proposed method for an arbitrary complete orthonormal systems of functions in the space $L_2([t, T]^k),$ $k\in\mathbb{N}$ as well as for complete orthonormal with weight $r(t_1)\ldots r(t_k)$ systems of functions in the space $L_2([t, T]^k),$ $k\in\mathbb{N}$. The comparison of the considered method with the well-known expansions of iterated Ito stochastic integrals based on the Ito formula and Hermite polynomials is given. The convergence in the mean of degree $2n$ $(n \in \mathbb{N})$ and with probability 1 of the proposed method is proved.