Mean-Square Approximation of Iterated Stochastic Integrals from Strong Exponential Milstein and Wagner-Platen Methods for Non-commutative Semilinear SPDEs Based on Multiple Fourier-Legendre Series

Abstract

AbstractThis work is devoted to the mean-square approximation of iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process. These integrals are part of the high-order strong numerical methods (with respect to the temporal discretization) for semilinear stochastic partial differential equations with nonlinear multiplicative trace class noise, which are based on the Taylor formula in Banach spaces and exponential formula for the mild solution of semilinear stochastic partial differential equations. For the approximation of the mentioned stochastic integrals we use the multiple Fourier–Legendre series converging in the sense of norm in Hilbert space. In this article, we propose the optimization of the mean-square approximation procedures for iterated stochastic integrals of multiplicities 1 to 3 with respect to the infnite-dimensional Q-Wiener process.KeywordsSemilinear stochastic partial differential equationInfinite-dimensional Q-Wiener processNonlinear multiplicative trace class noiseIterated stochastic integralGeneralized multiple Fourier seriesMultiple Fourier–Legendre seriesExponential Milstein schemeExponential Wagner–Platen schemeLegendre polynomialMean-square approximationExpansion