Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg–de Vries equations

Abstract

We establish weak convergence rates for noise discretizations of a wide class of stochastic evolution equations with non-regularizing semigroups and additive or multiplicative noise. This class covers the nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg-de Vries equation. For several important equations, including the stochastic wave equation, previous methods give only suboptimal rates, whereas our rates are essentially sharp. * We are grateful to Arnulf Jentzen for pointing us to the topic and sharing with us ideas from unpublished joint work with Sonja Cox [5]. M. S. M. thankfully acknowledges support by the Swiss National Science Foundation through grant SNF 205121 163425. P. H. thankfully acknowledges support by the Freiburg Institute of Advanced Studies in the form of a Junior Fellowship.

Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

“…Nonetheless, rigorous examination of simpler model problems such as the ones considered in the present work is a key first step. Even though a number of strong convergence rates for stochastic wave equations are available (cf., e.g., [3,7,8,30,39,44,45,48]), apart from the findings of the works Harms & Müller [20] and Cox et al [11], which have appeared after the preprint [23] of the present article, the existing weak convergence results for stochastic wave equations in the literature (cf., e.g., [22,28,29,31,45,46]) assume that the diffusion coefficient is constant or, in other words, that the equation is driven by additive noise.…”

“…However, the situation is different in the case of the infinitedimensional stochastic partial differential equations (cf., e.g., Walsh [43], Da Prato & Zabczyk [13], Liu & Röckner [35]). In the case of stochastic partial differential equations with regular nonlinearities, strong convergence rates are essentially well understood, whereas a proper understanding of weak convergence rates has still not been reached (cf., e.g., [1,2,[4][5][6]9,11,[16][17][18][19][20][21][22]25,[27][28][29]31,32,34,41,[45][46][47] for several weak convergence results in the literature). In this work we derive weak convergence rates for stochastic wave equations.…”

Numerical approximation and simulation of the stochastic wave equation on the sphere

“…We end this section by observing that several strategies for proving weak rates of convergence of numerical solutions to SPDEs in the literature could be extended to the present setting or in the case of numerical discretizations of nonlinear stochastic wave equations on the sphere, see for instance [13,24,34,5,20,16,23] and references therein. This could be subject of future research.…”

Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations

“…In addition, longtime behaviors of the numerical solutions of a linear model is investigated. The paper [32] provides a convergence rate of the weak error under noise discretizations of some Schrödinger's equations. Finally, the work [33] shows convergence in probability of a stochastic (implicit) symplectic scheme for stochastic nonlinear Schrödinger equations with quadratic potential and an additive noise.…”