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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:413

Article

Kovacs, Mihaly¹  Printems, Jacques² 

[1] Univ Otago, Dept Math & Stat, Dunedin 9054, New Zealand

[2] Univ Paris Est, Lab Anal & Math Appl, CNRS UMR 8050, F-94010 Creteil, France

关键词: Voltorra equations;    Euler scheme;    Stochastic partial differential equations;    Finite element method;    Weak convergence;    Convolution quadrature;    Fractional differential equations;   

DOI  :  10.1016/j.jmaa.2013.12.034

来源: Elsevier

【 摘 要 】

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 Ito form as dX(t) + (integral(t)(0) b(t - s)AX(s)ds) dt = dW(Q)(t), t is an element of (0, T]; X(0) = X-0 is an element of H, 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 is an element of (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((nu-1/rho)/2)Q(1/2) is a Hilbert-Schmidt operator on H for some nu is an element of [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). We show that for phi : H -> R twice continuously differentiable test function with bounded second derivative, vertical bar E phi(X-h(N)) - E phi(X(T))vertical bar <= C ln(T/h(2/rho) +Delta t) (Delta t(rho nu) + h(2 nu)), for any 0 <= nu <= 1/rho. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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