High order weak methods for stochastic differential equations based on modified equations

Abstract

Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new high order weak methods, in particular, implicit integrators well suited for stiff stochastic problems, and integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of the methodology.