A Stochastic Differential Equation Cookbook

Abstract

Abstract This chapter provides a tutorial on stochastic differential equations (SDEs). It starts by suggesting that, if one has a first-order differential equation dx/dt = a(x) + b(x)η(t), where η(t) is a randomly wildly fluctuating function of time, one may wish to approach its solution approximately by using SDE techniques. One starts with the approximation of replacing η(t) by white noise w(t). This then becomes a Stratonovich SDE. The chapter then provides a series of steps, a “recipe,” to obtain a satisfactory resolution of the problem. It explains how to add an “Ito correction term” to the Stratonovich SDE so that it becomes an Ito SDE. The next step is to (easily) pluck, from the Ito SDE, the ensemble mean values of dx/dt and (dx)2/dt (expressed in terms of a, b).The final step is to insert these expressions in the Fokker–Planck equation for ρ(x,t), the probability (density) of x at time t (whose solution gives one what is desired for the original first-order differential equation). Some examples of how this recipe works are provided. So far, as befits a recipe, no justification for these steps was provided. So, the chapter next justifies these steps. It then goes on to treat the general case of a set of first order differential equations, for a set of variables xi and a set of white noise functions wi(t).