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.