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 $\{u(t){\}}_{t\in [0,T]}$ 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 $$\text{dd}u+({\int }_0^{t}b(t-s)Au(s)\text{dd}s)\text{dd}t=\text{dd}{W}^{{}_Q},t\in (0,T];u(0)=u_0\in H,$$ 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 $$b(t)=t^{\beta -1}\mathrm{/}\mathrm{\Gamma }(\beta ),0<\beta <1.$$ 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 $\alpha -\frac{1}{\beta +1}<\kappa \le \alpha$. The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter $\mathrm{\Delta }t=T\mathrm{/}n$), and a standard continuous finite element method in space (parameter $h$). Let $u_{n,h}$ be the discrete solution at $T=n\mathrm{\Delta }t$. We show that $$(\text{E}\mathrm{\parallel }{u}_{n,h}-u(T){\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}=\mathcal{O}(h^{\nu }+\mathrm{\Delta }{t}^{\gamma }),$$ for any $\gamma <(1-(\beta +1)(\alpha -\kappa ))\mathrm{/}2$ and $\nu \le \frac{1}{\beta +1}-\alpha +\kappa$.