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

TL;DR

This work develops a general, mean-square convergent expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ using generalized multiple Fourier series with respect to an arbitrary complete orthonormal system in $L_2([t, T])$, extended to weight-based bases. The main result, Theorem 1, expresses $J[eta^{(k)}]_{T,t}$ as a limit of a $k$-fold Fourier-sum in Gaussian variables $oldsymboleta_j^{(i)}$ with a permutation-based correction, plus a controllable mean-square remainder; convergence is established in mean-square and, under smoothness conditions, almost surely. The paper further extends the framework to weighted orthonormal systems, analyzes discontinuous bases, compares to Hermite-polynomial representations, and connects the approach to Wong–Zakai approximations of Stratonovich integrals, including concrete examples and interval [t, s] modifications. Collectively, these results provide a versatile, analytically tractable toolkit for mean-square approximations of high-multiplicity stochastic integrals with potential impact on numerical SDE solvers and stochastic analysis. The framework emphasizes explicit coefficient structures, Wick-polynomial interpretations, and robust convergence guarantees across a broad class of bases and weights.

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.