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 X_{t}.
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 } - W_{t} 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 X_{t} .
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 X_{t}, are estimated by repeating this procedure as many times as necessary.