Cubature on Wiener space

Abstract

It is well known that there is a mathematical equivalence between ‘solving’ parabolic partial differential equations (PDEs) and ‘the integration’ of certain functionals on Wiener space. Monte Carlo simulation of stochastic differential equations (SDEs) is a naive approach based on this underlying principle. In finite dimensions, it is well known that cubature can be a very effective approach to integration. We discuss the appropriate extension of this idea to Wiener space. In the process we develop high–order numerical schemes valid for high–dimensional SDEs and semi–elliptic PDEs.