Cubature on Wiener space in infinite dimension

Abstract

We prove a stochastic Taylor expansion for stochastic partial differential equations (SPDEs) and apply this result to obtain cubature methods, i.e. high-order weak approximation schemes for SPDEs, in the spirit of Lyons and Victoir (Lyons & Victoir 2004 Proc. R. Soc. A 460 , 169–198). 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 Lévy processes of finite type; here the results seem to be new, even in finite dimension. Several numerical examples are added.