The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof

TL;DR

This paper reviews over thirty theorems on the expansion of iterated Ito and Stratonovich stochastic integrals, introduces new theorems and hypotheses, and discusses convergence properties relevant for numerical solutions of stochastic differential equations.

Contribution

It presents new theorems on the expansion of iterated Ito and Stratonovich stochastic integrals of arbitrary multiplicity, extending previous results and formulating hypotheses for further research.

Findings

Theorems on expansion of iterated Ito integrals based on Fourier series.

Adaptation of these theorems to Stratonovich integrals for multiplicities 1 to 8.

Convergence of expansions in the mean-square sense with a single limit operation.

Abstract

In this article, we collected more than thirty theorems on expansions of iterated Ito and Stratonovich stochastic integrals, which have been formulated and proved by the author in the period from 1997 to 2025. These theorems open up a new direction for study of iterated Ito and Stratonovich stochastic integrals. We consider two theorems on expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k)$. We adapt these theorems on expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ for iterated Stratonovich stochastic integrals of multiplicities 1 to 8 (the case of continuously differentiable weight functions and a complete orthonormal system of Legendre polynomials or trigonometric functions in $L_2([t, T])$) and for iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case of an arbitrary complete orthonormal system of functions in $L_2([t, T])$). On the base of the presented theorems we formulate several hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$. Recently, Hypothesis 8 has been proved for the case of an arbitrary complete orthonormal system of functions in $L_2([t, T]),$ $p_1=\ldots=p_k=p,$ $k\in\mathbb{N}$ but under one additional condition (Theorems 57, 59). The considered expansions converge in the mean-square sense and contain only one operation of the limit transition in contrast to its existing analogues. The mentioned iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion, i.e. the results of the article can be applied to numerical integration of Ito SDEs.