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 is an element of[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 du + (integral(t)(0) b(t - s)Au(s)ds) dt = dW(Q), t is an element of (0, T]; u(0) = u(0) is an element of H, where W-Q is a Q-Wiener process on H = L-2(D) and where the main example of b we consider is given by b(t) = t beta-1/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(k)) for some real kappa with alpha - 1/beta+1 < kappa <= 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 (E vertical bar vertical bar(un,h) - u(T)vertical bar vertical bar(2))(1/2) = O(h(v) + Delta t(gamma)), for any gamma < (1 - (beta + 1)(alpha - kappa))/2 and v <= 1/beta+1 - alpha+kappa