There is an explicit and an implicit version of the Milstein scheme.

The Milstein scheme is the simplest scheme that achieves a higher strong order of convergence than the Euler scheme, namely 1.0.

Details

Let

$d X = a \; d t + b \; d W$

be an Itô stochastic differential equation with $a, b$ suitable functions. In one dimension, the Milstein scheme is the following approximation scheme: