# Contents

## Idea

The Milstein scheme, named after Grigori N. Milstein, is a numerical discrete approximation scheme for stochastic differential equations.

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:

$Y_{n+1} = Y_n + \alpha \; a(\tau_{n+1}, Y_{n+1}) (\tau_{n+1} - \tau_{n}) + (1 - \alpha) a(\tau_{n}, Y_{n}) (\tau_{n+1} - \tau_{n}) + b (W_{\tau_{n+1}} - W_{\tau_n}) + \frac{1}{2} b b' ( (W_{\tau_{n+1}} - W_{\tau_n} )^2 - (\tau_{n+1} - \tau_{n}))$

The number $\alpha$ is chosen to be between 0 and 1.