Stochastic calculus

A stochastic differential equation (SDE) is written in the following form:

 equation209

If tex2html_wrap_inline29 (x,t) is always zero then the equation is just an ordinary differential equation;
if tex2html_wrap_inline29 (x,t) is independent of x then (1) is a differential equation with additive white noise.
Here X is a stochastic process; its value at time t is a random variable denoted by Xt.

The Wiener process W , also called standard Brownian motion, satisfies (1) with f(x,t)=0 and tex2html_wrap_inline29(x,t)=1 . Its paths have the property that, for any t and tex2html_wrap_inline29t, the random variable W t+ tex2html_wrap_inline29t - Wt is Gaussian with mean 0 and variance tex2html_wrap_inline29t, and successive increments are independent.

Solving the SDE (1) on a computer consists of generating a set of values: { X(); i=1,..., N } , where 0<t_1<t_2<...< t_N=t , that approximate the corresponding values of Xt . In the lowest order algorithm X() is generated from X() by adding a deterministic increment and a random one:
 equation226

where tex2html_wrap_inline29t= - and each n i is independently generated from a Gaussian density with mean zero and variance 1. Properties of the ensemble of paths, such as the mean value of Xt, are estimated by repeating this procedure as many times as necessary.


  • Notation and definitions
  • What is dw?
  • References