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 $$b(t)=t^{{\hat{I}}^2-1}/\hat{I}“({\hat{I}}^2),0<{\hat{I}}^2<1.$$ We let $A$ be an unbounded linear self-adjoint positive operator on $H$ and we further assume that there exist $\hat{I}\pm >0$ such that $A^{-\hat{I}\pm }$ has finite trace and that $Q$ is bounded from $H$ into $D(A^{\hat{I}}º)$ for some real $\hat{I}º$ with $\hat{I}\pm -\frac{1}{{\hat{I}}^2+1}<\hat{I}º\le \hat{I}\pm$. The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter $\hat{I}”t=T/n$), and a standard continuous finite element method in space (parameter $h$). Let $u_{n,h}$ be the discrete solution at $T=n\hat{I}”t$. We show that $$ (\E \| u_{n,h} - u(T)\|^2)^{1/2}={\mathcal O}(h^ν + Δt^γ), $$ for any ${\hat{I}}^3<(1-({\hat{I}}^2+1)(\hat{I}\pm -\hat{I}º))/2$ and $\hat{I}½\le \frac{1}{{\hat{I}}^2+1}-\hat{I}\pm +\hat{I}º$.

To appear in Math. Comp