Weak order for the discretization of the stochastic heat equation driven by impulsive noise

Abstract

Considering a linear parabolic stochastic partial differential equation driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H, t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an impulsive cylindrical process and Q describes the spatial covariance structure of the noise; Tr(A^{-α})<\infty for some α>0 and A^βQ is bounded for some β\in(α-1,α]. A discretization (X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space (parameter h>0) and a θ-method in time (parameter Δt=T/N). For ϕ\in C^2_b(H;R) we show an integral representation for the error |Eϕ(X^N_h)-Eϕ(X_T)| and prove that |Eϕ(X^N_h)-Eϕ(X_T)|=O(h^{2γ}+(Δt)^γ) where γ<1-α+β.