Milstein scheme (changes)

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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.

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:

$${Y}_{n+1}={Y}_{n}+\alpha \phantom{\rule{thickmathspace}{0ex}}a({\tau}_{n+1},{Y}_{n+1})({\tau}_{n+1}-{\tau}_{n})+(1-\alpha )a({\tau}_{n},{Y}_{n})({\tau}_{n+1}-{\tau}_{n})+b({W}_{{\tau}_{n+1}}-{W}_{{\tau}_{n}})+\frac{1}{2}bb\prime (({W}_{{\tau}_{n+1}}-{W}_{{\tau}_{n}}{)}^{2}-({\tau}_{n+1}-{\tau}_{n}))$$ 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.

- Milstein method, Wikipedia.

category: computational methods