Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise

Abstract

In numerical analysis for stochastic partial differential equations one distinguishes between weak and strong convergence rates. Often the weak convergence rate is twice the strong convergence rate. However, there is no standard way to prove this: to obtain optimal weak convergence rates for stochastic partial differential equations requires specially tailored techniques, especially if the noise is multiplicative. In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. The weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild Itô formula.

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

“…� , [5,11,14,15,18,20,23,26,27,38] and references therein. This could be subject of future research.…”

Cylindrical stochastic integration

“…Our modeling framework is versatile and can be applied to other financial markets, as well, including energy markets. In principle, it also lends itself to numerical discretization by similar methods as for non-cylindrical models [1,12], but we have not explored this any further.…”

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

“…In order to implement an approximation on a computer, the equation is typically discretized both in the spatial and temporal parameters, in which case the resulting approximation û is said to be fully discrete. In the literature, the quality of û is in general evaluated by analyzing the rate of decay of the strong error E[ u− û 2 L 2 (D) ] 1/2 (see [3,6,7,8,9,15,16,17,18,23,25,26,27]). Comparatively few results (see [9,14,15,16,17,18,26]) exist on the rate for the weak error E[|φ(u) − φ(û)|], where φ : L 2 (D) → R is a sufficiently smooth real-valued test function.…”

“…In the literature, the quality of û is in general evaluated by analyzing the rate of decay of the strong error E[ u− û 2 L 2 (D) ] 1/2 (see [3,6,7,8,9,15,16,17,18,23,25,26,27]). Comparatively few results (see [9,14,15,16,17,18,26]) exist on the rate for the weak error E[|φ(u) − φ(û)|], where φ : L 2 (D) → R is a sufficiently smooth real-valued test function. Of the results cited, only [14] provides a weak convergence result for a fully discrete approximation of a semilinear stochastic wave equation.…”

“…In Section 3 we deduce strong and weak error rates for the approximation of the so called mild solution u of (1) by means of a finite element approximation (by piecewise linear or quadratic functions) in space and a rational approximation of the exponential function in time, generalizing the result of [17] to the semilinear setting. This approach sets the paper apart from several recent works (e.g., [3,7,8,9,26,27]) on the stochastic wave equation that consider trigonometric integrators for the temporal approximation. There are situations when such integrators could be better suited such as highly oscillatory data but for complicated domain geometries the algorithms in the present article could be more advantageous from an implementation point of view, since they not require any knowledge of the eigenfunctions of ∆ or its discrete counterpart.…”