Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation

Abstract

In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process $\{u(t)\}_{t\in [0,T]}$ satisfying a stochastic linear evolution equation with a positive-type memory term driven by an additive Gaussian noise. The equation can be written in an abstract form as $$ \dd u + (\int_0^t b(t-s) Au(s) \, \dd s)\, \dd t = \dd W^{_Q}, t\in (0,T]; \quad u(0)=u_0 \in H, $$ where $W^{_Q}$ is a $Q$-Wiener process on $H=L^2({\mathcal D})$ and where the main example of $b$ we consider is given by $$ b(t) = t^{β-1}/Γ(β), \quad 0 < β<1. $$ We let $A$ be an unbounded linear self-adjoint positive operator on $H$ and we further assume that there exist $α>0$ such that $A^{-α}$ has finite trace and that $Q$ is bounded from $H$ into $D(A^κ)$ for some real $κ$ with $α-\frac{1}{β+1}<κ\leq α$. The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter $Δt =T/n$), and a standard continuous finite element method in space (parameter $h$). Let $u_{n,h}$ be the discrete solution at $T=nΔt$. We show that $$ (\E \| u_{n,h} - u(T)\|^2)^{1/2}={\mathcal O}(h^ν + Δt^γ), $$ for any $γ< (1 - (β+1)(α- κ))/2 $ and $ν\leq \frac{1}{β+1}-α+κ$.