Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs

TL;DR

The work develops a robust framework for the strong (mean-square) approximation of iterated Itô and Stratonovich stochastic integrals using generalized multiple Fourier series. It proves mean-square (Theorem 1.1) and mean-square-error (Theorem 1.3) expansions with explicit Fourier coefficients and Gaussian weights, valid for arbitrary multiplicity $k$ and flexible bases such as Legendre polynomials or trigonometric systems. The approach yields exact and computable expressions for the mean-square error, includes detailed expansions for low multiplicities ($k\le 7$) and a general construction for any $k$, and discusses extensions to discontinuous bases and Stratonovich integrals. These results support high-order numerical schemes for Itô SDEs and semilinear SPDEs by providing explicit, controllable approximations to the necessary iterated stochastic integrals, with insights into basis selection and computational efficiency. The methodology also interfaces with broader stochastic-analysis tools, including comparisons with Hermite-based representations and applications to Q-Wiener processes.

Abstract

The book is devoted to the strong approximation of iterated stochastic integrals (ISIs) in the context of numerical integration of Ito SDEs and non-commutative semilinear SPDEs with nonlinear multiplicative trace class noise. The monograph opens up a new direction in researching of ISIs. For the first time we successfully use the generalized multiple Fourier series converging in the sense of norm in $L_2([t, T]^k)$ for the expansion and strong approximation of Ito ISIs of multiplicity $k,$ $k\in{\bf N}$ (Chapter 1). This result has been adapted for Stratonovich ISIs of multiplicities 1 to 8 (the case of continuously differentiable weight functions and a CONS of Legendre polynomials or trigonometric functions in $L_2([t, T])$) and for Stratonovich ISIs of multiplicities 1 to 6 (the case of continuous weight functions and an arbitrary CONS in $L_2([t, T])$) (Chapter 2), as well as for some other types of iterated stochastic integrals (Chapter 1). Recently (in 2024), the mentioned adaptation has also been carried out for Stratonovich ISIs of multiplicity $k$ $(k\in{\bf N})$ for the case of an arbitrary CONS in $L_2([t, T])$ (Theorems 2.59, 2.61) but under one additional condition. We derived the exact and approximate expressions for the mean-square error of approximation of Ito ISIs of multiplicity $k$, $k\in{\bf N}$ (Chapter 1). We provided a significant practical material (Chapter 5) devoted to the expansions of specific Ito and Stratonovich ISIs of multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4) using the CONS of Legendre polynomials and the CONS of trigonometric functions. The methods formulated in this book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of ISIs of multiplicity $k,$ $k\in{\bf N}$ with respect to the $Q$-Wiener process.