Cubature on Wiener space in infinite dimension

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 Levy processes of finite type; here the results seem to be new, even infinite dimension. Several numerical examples are added.