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

In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process ${u(t)}_{tin [0,T]}$ 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 $$ dd u + (int_0^t b(t-s) Au(s) , dd s), dd t = dd W^{_Q}, tin (0,T]; quad u(0)=u_0 in H, $$ 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 $$ b(t) = t^{beta-1}/Gamma(beta), quad 0 < beta <1. $$ We let $A$ be an unbounded linear self-adjoint positive operator on $H$ and we further assume that there exist $alpha >0$ such that $A^{-alpha}$ has finite trace and that $Q$ is bounded from $H$ into $D(A^kappa)$ for some real $kappa$ with $alpha-frac{1}{beta+1}<kappa leq alpha$. The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter $Delta t =T/n$), and a standard continuous finite element method in space (parameter $h$). Let $u_{n,h}$ be the discrete solution at $T=nDelta t$. We show that $$ (E | u_{n,h} - u(T)|^2)^{1/2}={mathcal O}(h^{ u} + Delta t^gamma), $$ for any $gamma< (1 - (beta+1)(alpha - kappa))/2 $ and $ u leq frac{1}{beta+1}-alpha+kappa$.