Numerical Integration of Stochastic Differential Equations - II

There are two approaches to the study of a physical system described by a stochastic differential equation (SDE). On the one hand, one may work with an equation for the probability distribution function for the random variables such as the Fokker-Planck equation. On the other hand, one may attempt to generate representative points on a trajectory by direct solution of the SDE. With either approach it is rare that analytical solutions can be found, except for linear systems. While the deterministic equation for the probability distribution can be solved numerically with standard techniques, in practice there are great difficulties. Numerical techniques for S D E S are a less-developed subject, but quite promising since they are capable of giving direct information about the random process, such as the power spectrum, higher moments, and transition rates. Several discussions of the problem have been published.1 A previous paper2 (hereafter referred to as I) describes a systematic approach to the numerical solution of SDES. Attention was limited to the simple one-variable equation of the form 1927