Efficient computing of stochastic differential equations

Stochastic differential equations are more appropriate than deterministic differential equations in modeling a large class of physical problems. Although exact solutions are impossible to be obtained in the nonlinear multivariable case, numerical methods are necessary and significant to the development of science and technology, see [1] for an introduction. In this paper we developed a family of STPH schemes by means of Chen series in terms of Philip Hall basis such that the number of multiple stochastic integrals is greatly reduced especially in high order approximation. Derivative free schemes are obtained if partial derivatives are approximated by finite difference, see also [2].