Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term

Abstract

In this paper we are interested in the numerical approximation of the marginal distributions of the Hilbert space valued solution of a stochastic Volterra equation driven by an additive Gaussian noise. This equation can be written in the abstract Itô form asdX(t)+(∫0tb(t−s)AX(s)ds)dt=dWQ(t),t∈(0,T];X(0)=X0∈H, where WQ is a Q-Wiener process on the Hilbert space H and where the time kernel b is the locally integrable potential tρ−2, ρ∈(1,2), or slightly more general. The operator A is unbounded, linear, self-adjoint, and positive on H. Our main assumption concerning the noise term is that A(ν−1/ρ)/2Q1/2 is a Hilbert–Schmidt operator on H for some ν∈[0,1/ρ]. The numerical approximation is achieved via a standard continuous finite element method in space (parameter h) and an implicit Euler scheme and a Laplace convolution quadrature in time (parameter Δt=T/N). We show that for φ:H→R twice continuously differentiable test function with bounded second derivative,|Eφ(XhN)−Eφ(X(T))|⩽Cln(Th2/ρ+Δt)(Δtρν+h2ν), for any 0⩽ν⩽1/ρ. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.