Weak error analysis for semilinear stochastic Volterra equations with additive noise

TL;DR

This paper establishes a weak error estimate for numerical approximations of semilinear stochastic Volterra equations with additive noise, demonstrating that the weak convergence rate is twice the strong rate using Malliavin calculus.

Contribution

It introduces a novel weak error analysis for semilinear stochastic Volterra equations, including path-dependent functionals, without relying on Kolmogorov equations.

Findings

Weak convergence rate is twice the strong rate.

Convergence includes functionals of the entire path and higher order statistics.

Method applies to a broad class of stochastic integro-differential equations.

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.