Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs

Abstract

This article investigates the role of the regularity of the test function when considering the weak error for standard discretizations of SPDEs of the form dX(t) = AX(t)dt + F (X(t))dt + dW (t), driven by space-time white noise. In previous results, test functions are assumed (at least) of class C 2 with bounded derivatives, and the weak order is twice the strong order.We prove, in the case F = 0, that to quantify the speed of convergence, it is crucial to control some derivatives of the test functions, even when the noise is non-degenerate. First, the supremum of the weak error over all bounded continuous functions, which are bounded by 1, does not converge to 0 as the discretization parameter vanishes. Second, when considering bounded Lipschitz test functions, the weak order of convergence is divided by 2, i.e. it is not better than the strong order. This is in contrast with the finite dimensional case, where the Euler-Maruyama discretization of elliptic SDEs dY (t) = f (Y (t))dt + dB t has weak order of convergence 1 even for bounded continuous functions.1991 Mathematics Subject Classification. 60H15,60H35,65C30.

Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme

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Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

“…In the context of the numerical approximation of the solution processes of such equations, the quantity of interest is typically the expected value of some functional of the solution and one is thus interested in the weak convergence rate of the considered numerical scheme. While the weak convergence analysis for numerical approximations of SPDE with Gaussian noise is meanwhile relatively far developed, see, e.g., [1,2,3,7,8,9,10,11,12,14,15,16,17,18,23,30], available results for non-Gaussian Lévy noise have been restricted to linear equations so far [4,5,21,25]. In this article, we analyze for the first time the weak convergence rate of numerical approximations for a class of semi-linear SPDE with non-Gaussian Lévy noise.…”

Convergence of Density Approximations for Stochastic Heat Equation

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