Weak Convergence of a Fully Discrete Approximation of a Linear Stochastic Evolution Equation with a Positive-type Memory Term Mihály Kovács and Jacques Printems

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 as dX(t) + (∫ t 0 b(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 ∆t = T/N). We show that for φ : H → R twice continuously differentiable test function with bounded second derivative, |Eφ(X h )−Eφ(X(T ))| ≤ C ln ( T h2/ρ + ∆t ) (∆t + h), for any 0 ≤ ν ≤ 1/ρ. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.