Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise

Abstract

We study the approximation of the distribution of X T , where (X t ) t ∈ [0, T] is a Hilbert space valued stochastic process that solves a linear parabolic stochastic partial differential equation driven by an impulsive space time noise, $$ dX_t+AX_t\,dt= Q^{1/2}\,dZ_t,\quad X_0=x_0\in H,\quad t\in [0,T]. $$ Here (Z t ) t ∈ [0, T] is an impulsive cylindrical process and the operator Q describes the spatial covariance structure of the noise; we assume that A − α has finite trace for some α > 0 and that A β Q is bounded for some β ∈ (α − 1, α]. A discretized solution $(X_h^n)_{n\in\{0,1,\ldots,N\}}$ is defined via the finite element method in space (parameter h > 0) and a θ-method in time (parameter Δt = T/N). For $\varphi \in C^2_b(H;{\mathbb R})$ we show an integral representation for the error $|{\mathbb E}\varphi(X^N_h)-{\mathbb E}\varphi(X_T)|$ and prove that $$ \left|{\mathbb E}\varphi\left(X^N_h\right)-{\mathbb E}\varphi(X_T)\right|=O\left(h^{2\gamma}+\left(\Delta t\right)^{\gamma}\right) $$ where γ < 1 − α + β. This is the same order of convergence as in the case of a Gaussian space time noise, which has been obtained in a paper by Debussche and Printems (Math Comput 78:845–863, 2009). Our result also holds for a combination of impulsive and Gaussian space time noise.