Iterated Stochastic Integrals in Infinite Dimensions - Approximation and Error Estimates

TL;DR

This paper extends algorithms for approximating iterated stochastic integrals from finite to infinite dimensions, enabling advanced numerical schemes for SPDEs without commutative noise, with derived error estimates for these methods.

Contribution

It generalizes existing finite-dimensional algorithms to infinite-dimensional settings, providing new tools for simulating SPDEs with non-commutative noise.

Findings

Error estimates in mean-square sense for both algorithms.

Approximation algorithms depend on the covariance operator Q.

Extension of finite-dimensional methods to infinite-dimensional stochastic processes.

Abstract

Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden, Platen, and Wright (1992) and by Wiktorsson (2001) for the approximation of two-times iterated stochastic integrals involved in numerical schemes for finite dimensional stochastic ordinary differential equations to an infinite dimensional setting. These methods clear the way for new types of approximation schemes for SPDEs without commutative noise. Precisely, we analyze two algorithms to approximate two-times iterated integrals with respect to an infinite dimensional $Q$-Wiener process in case of a trace class operator $Q$ given the increments of the $Q$-Wiener process. Error estimates in the mean-square sense are derived and discussed for both methods. In contrast to the finite dimensional setting, which is contained as a special case, the optimal approximation algorithm cannot be uniquely determined but is dependent on the covariance operator $Q$. This difference arises as the stochastic process is of infinite dimension.