Weak error analysis for semilinear stochastic Volterra equations with additive noise

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:437

Article

Andersson, Adam¹  Kovacs, Mihaly²  Larsson, Stig^3,4 

[1] Tech Univ Berlin, Inst Math, Secr MA 5-3,Str 17 Juni 136, DE-10623 Berlin, Germany

[2] Univ Otago, Dept Math & Stat, POB 56, Dunedin 9054, New Zealand

[3] Chalmers, Dept Math Sci, SE-41296 Gothenburg, Sweden

[4] Univ Gothenburg, SE-41296 Gothenburg, Sweden

关键词: Stochastic Volterra equations;    Finite element method;    Backward Euler;    Convolution quadrature;    Strong and weak convergence;    Malliavin regularity;   

DOI  :  10.1016/j.jmaa.2015.09.016

来源: Elsevier

【 摘 要 】

We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus. (C) 2016 Published by Elsevier Inc.

【 授权许可】

Free