The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series

TL;DR

The work develops a rigorous almost sure convergence theory for expansions of iterated Itô stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series, with norm convergence in $L_2([t, T]^k)$. It represents the kernel $K(t_1,\dots,t_k)$ in a generalized Fourier basis, yielding a mean-square convergent expansion in terms of Gaussian variables $\zeta_j^{(i)}$ and Fourier coefficients $C_{j_k\ldots j_1}$, and provides explicit error bounds tied to Parseval-type sums. The central result, Theorem 6, shows $w.p.1$ convergence of the truncated expansions for continuously differentiable $\psi_l$ using Legendre or trig bases, supported by moment estimates and Parseval/Dini arguments; the Appendix supplies the necessary technical lemmas. Collectively, the paper equips high-order stochastic numerical methods for Itô SDEs with explicit, verifiable expansion formulas and convergence guarantees, including applicability to infinite-dimensional $Q$-Wiener processes via flexible orthonormal bases.

Abstract

The article is devoted to the formulation and proof of the theorem on convergence with probability 1 of expansion of iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series converging in the sense of norm in Hilbert space. The cases of multiple Fourier-Legendre series and multiple trigonomertic Fourier series are considered in detail. The proof of the mentioned theorem is based on the general properties of multiple Fourier series as well as on the estimate for the fourth moment of approximation error in the method of expansion of iterated Ito stochastic integrals based on generalized multiple Fourier series.