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

TL;DR

This paper studies the numerical approximation of the distribution of solutions to a stochastic Volterra equation with memory, using finite element and convolution quadrature methods, and establishes convergence rates.

Contribution

It provides the first convergence analysis for fully discrete schemes approximating the marginal distributions of such stochastic Volterra equations with positive-type memory.

Findings

Convergence rate of order \\Delta t^{\rho \nu} + h^{2\nu} for the approximation.

Logarithmic error bound involving spatial and temporal discretization parameters.

Extension of strong convergence results to distributional convergence for the numerical scheme.

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 It\^o 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 $t^{\rho-2}$, $\rho \in (1,2)$, 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 $A^{(\nu- 1/\rho)/2} Q^{1/2}$ is a Hilbert-Schmidt operator on $H$ for some $\nu \in [0,1/\rho]$. 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 $\Delta t=T/N$). %Let $X_h^N$ be the discrete solution at time $T$. Eventually let $\varphi : H\rightarrow \R$ is such that $D^2\varphi$ is bounded on $H$ but not necessarily bounded and suppose in addition that either its first derivative is bounded on $H$ and $X_0 \in L^1(\Omega)$ or $\varphi = \| \cdot \|^2$ and $X_0 \in L^2(\Omega)$. We show that for $\varphi : H\rightarrow \R$ twice continuously differentiable test function with bounded second derivative, $$ | \E \varphi(X^N_h) - \E \varphi(X(T)) | \leq C \ln \left(\frac{T}{h^{2/\rho} + \Delta t} \right) (\Delta t^{\rho \nu} + h^{2\nu}), $$ \noindent for any $0\leq \nu \leq 1/\rho$. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.