Milstein method

The Milstein method of stochastic analysis is a method for the numerical solution of stochastic differential equations ( SDE ), named after the Russian mathematician Grigori Noichowitsch Milstein (State Gorky University of the Ural ).

Algorithm

Consider the Ito - SDE

With initial condition, where the Wiener process referred. If a solution on the interval be found, obtained by the Milstein method, an approximation to the true solution on an equidistant grid:

• Decompose the interval into equal subintervals of length:

• Set.

• Define for by

In which

And the derivative of with respect to is. Note that the random variables are independently distributed normal with mean 0 and variance.

Convergence

With the above notation applies for and all, which is why we speak of convergence of the first order. is a Landau symbol.