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 as $$ \dd X(t) + \left (\int_0^t b(t-s) A X(s) \, \dd s \right) \, \dd t = \dd W^{_Q}(t), t\in (0,T]; ~ X(0) =X_0\in H, $$ \noindent where $W^Q$ is a $Q$-Wiener process on the Hilbert space $H$ and where the time kernel $b$ is the locally integrable potential $t^{\ddot{I}-2}$, $\ddot{I}\in (1,2)$, 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 $A^{(\hat{I}½-1/\ddot{I})/2}{Q}^{1/2}$ is a Hilbert-Schmidt operator on $H$ for some $\hat{I}½\in [0,1/\ddot{I}]$. 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 $\hat{I}”t=T/N$). %Let $X_h^N$ be the discrete solution at time $T$. Eventually let $\ddot{I}†:H\to \mathbb{R}$ is such that $D^2\ddot{I}†$ is bounded on $H$ but not necessarily bounded and suppose in addition that either its first derivative is bounded on $H$ and $X_0\in L^1(\hat{I}\text{c◯})$ or $\ddot{I}†=\Vert \cdot {\Vert }^2$ and $X_0\in L^2(\hat{I}\text{c◯})$. We show that for $\ddot{I}†:H\to \mathbb{R}$ twice continuously differentiable test function with bounded second derivative, $$ | \E φ(X^N_h) - \E φ(X(T)) | \leq C \ln \left(\frac{T}{h^{2/ρ} + Δt} \right) (Δt^{ρν} + h^{2ν}), $$ \noindent for any $0\le \hat{I}½\le 1/\ddot{I}$. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.