The Magnus Expansion for Stochastic Differential Equations

In this paper, all the terms in the stochastic Magnus expansion are presented by rooted trees. First, stochastic Magnus methods for linear stochastic differential equations are constructed by truncating the stochastic Magnus expansion. Then, explicit stochastic Magnus methods are constructed by Picard’s iteration for nonlinear stochastic differential equations on matrix Lie group. Furthermore, general nonlinear stochastic differential equations are transformed into linear operator stochastic differential equations by using the Lie derivative. Finally, numerical methods for general nonlinear stochastic differential equations are constructed by using the theory of the stochastic Magnus expansion for the linear case. In particular, for the commutative case, it is shown that the stochastic Magnus expansion provides a novel way to construct computationally inexpensive and arbitrarily high-order numerical methods while avoiding the simulation of multiple stochastic integrals. Moreover, the proposed methods are shown to preserve the intrinsic properties of the original system well and the numerical experiments agree with the theoretical results.