# Stochastic calculus

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

If (x,t) is always zero then the equation is just an ordinary differential equation;
if (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 (x,t)=1 . Its paths have the property that, for any t and t, the random variable W t+ t - Wt is Gaussian with mean 0 and variance t, 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:

where t= - 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