Consider the problem of numerically integrating the Itô stochastic differential equation,
where is a vector of dimension n, is a vector function of dimension n, is a k-dimensional Wiener process and is an matrix valued function.
Milshtein (1978) derived the following scheme for numerically solving (1),
( is a Gaussian random variable with zero mean and unit variance).
This scheme is , and is .
The above scheme is not used in the implementation of software to do the calculation because of the difficuties the high derivatives that need to be evaluated at each step. Therefore for numerical use, the following scheme is used,
Milshtein, shows that (3) is equivalent to (2).