A new proof of the expansion of iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite polynomials

TL;DR

This paper presents a new, more general proof of the expansion of iterated Ito stochastic integrals using Hermite polynomials and Fourier series, enabling advanced numerical methods for stochastic differential equations.

Contribution

The paper generalizes previous expansions of iterated Ito integrals to arbitrary orthonormal systems in Hilbert spaces using a novel approach based on Ito's formula.

Findings

Provides a more general expansion applicable to any complete orthonormal system.

Facilitates the development of high-order numerical methods for stochastic differential equations.

Uses a new proof technique based on Ito's formula.

Abstract

The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality. Namely, for the case of continuous or piecewise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.