Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term

In this paper we are interested in the numerical approximation of the marginal distributions of the Hilbert space valued solution of a stochastic Volterra equation driven by an additive Gaussian noise. This equation can be written in the abstract It^o form as $$ dd X(t) + left (int_0^t b(t-s) A X(s) , dd s right) , dd t = dd W^{_Q}(t), tin (0,T]; ~ X(0) =X_0in H, $$ oindent where $W^Q$ is a $Q$-Wiener process on the Hilbert space $H$ and where the time kernel $b$ is the locally integrable potential $t^{rho-2}$, $rho in (1,2)$, or slightly more general. The operator $A$ is unbounded, linear, self-adjoint, and positive on $H$. Our main assumption concerning the noise term is that $A^{( u- 1/rho)/2} Q^{1/2}$ is a Hilbert-Schmidt operator on $H$ for some $ u in [0,1/rho]$. The numerical approximation is achieved via a standard continuous finite element method in space (parameter $h$) and an implicit Euler scheme and a Laplace convolution quadrature in time (parameter $Delta t=T/N$). %Let $X_h^N$ be the discrete solution at time $T$. Eventually let $varphi : Hrightarrow R$ is such that $D^2varphi$ is bounded on $H$ but not necessarily bounded and suppose in addition that either its first derivative is bounded on $H$ and $X_0 in L^1(Omega)$ or $varphi = | cdot |^2$ and $X_0 in L^2(Omega)$. We show that for $varphi : Hrightarrow R$ twice continuously differentiable test function with bounded second derivative, $$ | E varphi(X^N_h) - E varphi(X(T)) | leq C ln left(frac{T}{h^{2/rho} + Delta t} right) (Delta t^{rho u} + h^{2 u}), $$ oindent for any $0leq u leq 1/rho$. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.

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$.