Weak error analysis for semilinear stochastic Volterra equations with additive noise
Journal article, 2016

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.

Stochastic Volterra equation

Finite element method

Malliavin calculus

Convolution quadrature

Duality

Strong and weak convergence

Regularity

Backward Euler

Author

Adam Andersson

Technische Universität Berlin

Mihaly Kovacs

University of Otago

Stig Larsson

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Journal of Mathematical Analysis and Applications

0022-247X (ISSN) 1096-0813 (eISSN)

Vol. 437 2 1283-1304

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Roots

Basic sciences

DOI

10.1016/j.jmaa.2015.09.016

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3/1/2018 1