Cubature on Wiener space in infinite dimension

TL;DR

This paper develops high-order weak approximation schemes for stochastic partial differential equations using cubature methods on Wiener space, extending to Lévy processes and providing numerical examples.

Contribution

It introduces a stochastic Taylor expansion for SPDEs and applies it to create cubature methods with high-order convergence, including novel results for Lévy processes.

Findings

High-order weak convergence for certain test functions

Extension of cubature methods to Lévy processes

Numerical examples demonstrating effectiveness

Abstract

We prove a stochastic Taylor expansion for SPDEs and apply this result to obtain cubature methods, i. e. high order weak approximation schemes for SPDEs, in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak convergence for well-defined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of L\'evy processes of finite type, here the results seem to be new even in finite dimension. Several numerical examples are added.