Order Conditions of Stochastic Runge--Kutta Methods by B-Series

In this paper, general order conditions and a global convergence proof are given for stochastic Runge--Kutta methods applied to stochastic ordinary differential equations (SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.