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 \[ \mathrm {d} u + \left ( \int _0^t b(t-s) Au(s) \mathrm {d} s \right ) \mathrm {d} t = \mathrm {d} W^{_Q},~t\in (0,T]; \quad u(0)=u_0 \in H, \] 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 \[ b(t) = t^{\beta -1}/\Gamma (\beta ), \quad 0 < \beta <1. \] 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 $T=n\Delta t$. We show that \begin{equation*} \left ( \mathbb {E} \| u_{n,h} - u(T)\|^2 \right )^{1/2}={\mathcal O}(h^{\nu } + \Delta t^\gamma ), \end{equation*} for any $\gamma < (1 - (\beta +1)(\alpha - \kappa ))/2$ and $\nu \leq \frac {1}{\beta +1}-\alpha +\kappa$.