A Lie Algebraic Approach to Numerical Integration of Stochastic Differential Equations

Abstract

In this article, "composition methods (or operator splitting methods)" for autonomous stochastic differential equations (SDEs) are formulated to produce numerical approximation schemes for the equations. In the proposed methods, the exponential map, which is given by the solution of an SDE, is approximated by composition of the stochastic flows derived from simpler and exactly integrable vector field operators having stochastic coefficients. The local and global errors of the numerical schemes derived from the stochastic composition methods are investigated. The new schemes are advantageous to preserve the special character of SDEs numerically and are useful for approximations of the solutions to stochastic nonlinear equations. To examine their superiority, several numerical simulations on the basis of the proposed schemes are carried out for SDEs which arise in mathematical finance and stochastic Hamiltonian dynamical systems.