Malliavin regularity and weak approximation of semilinear SPDE with L\'evy noise

TL;DR

This paper studies the weak convergence rates of numerical approximations for semilinear parabolic SPDEs driven by Lévy noise, extending Malliavin calculus techniques to a Poissonian setting.

Contribution

It introduces a novel analysis of Malliavin regularity for solutions of SPDEs with Lévy noise and establishes weak convergence rates that are twice the strong rates.

Findings

Weak convergence rate is twice the strong rate for certain test functions.

Extended Malliavin-Sobolev space results from Gaussian to Lévy (Poissonian) setting.

Provides theoretical foundation for numerical schemes of SPDEs with Lévy noise.

Abstract

We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable L\'evy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.