Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients

TL;DR

This paper establishes sharp weak convergence rates for Euler-type numerical approximations of semilinear stochastic evolution equations with nonlinear diffusion, filling a significant gap in the understanding of such methods.

Contribution

The article provides the first essentially sharp weak convergence rates for Euler-type methods applied to semilinear SEEs with nonlinear diffusion coefficients.

Findings

Sharp weak convergence rates are proven for linear-implicit Euler approximations.

The approach uses a mild Itô formula and semilinear integrated processes.

Results extend understanding of numerical methods for nonlinear SPDEs.

Abstract

Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have been investigated since about 12 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In the recent article [D. Conus, A. Jentzen & R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, arXiv:1408.1108] the weak convergence problem emerged from Debussche's article has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome the problem emerged from Debussche's article in the case of a class of time-discrete Euler-type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild It\^o type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.