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.