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arXiv · 1411.6476

Weak error analysis for semilinear stochastic Volterra equations with additive noise

Adam Andersson·Mihály Kovács·Stig Larsson
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Abstract

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.

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Adam Andersson, Mihály Kovács, Stig Larsson. 2014-11-24. Weak error analysis for semilinear stochastic Volterra equations with additive noise. https://doi.org/10.1016/j.jmaa.2015.09.016

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