Discretization of stochastic differential equations by the product expansion for the chen series

Abstract

Following the approach of Sussmann [16] to expand the Chen series as a product of an exponential Lie series based on a Philip Hall basis, this paper constructs the Stratonovich-Taylor-Hall (STH) discretization schemes of both integral and fractional orders for stochastic differential equations. The STH schemes are numerically efficient in the sense that they involve only the minimum numbers of multiple stochastic integrals. General multiple stochastic integrals are regarded as systems of stochastic differential equations that can be solved by lower order schemes. Convergence of the STH schemes is proved in the almost sure sense by means of the discrete stochastic Gronwall inequality