In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori N. Milstein who first published the method in 1974.
with initial condition X0 = x0, where Wt stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time [0, T]. Then the Milstein approximation to the true solution X is the Markov chain Y defined as follows:
- partition the interval [0, T] into N equal subintervals of width :
- recursively define for by
where denotes the derivative of with respect to and
For this derivation, we will only look at geometric Brownian motion (GBM), the stochastic differential equation of which is given by
with real constants and . Using Itō's lemma we get
Thus, the solution to the GBM SDE is
- Mil'shtein, G. N. (1974). "Approximate integration of stochastic differential equations". Teor. Veroyatnost. i Primenen (in Russian) 19 (3): 583–588.
- Mil’shtejn, G. N. (1975). "Approximate Integration of Stochastic Differential Equations". Theory of Probability & Its Applications 19 (3): 557–000. doi:10.1137/1119062.
- V. Mackevičius, Introduction to Stochastic Analysis, Wiley 2011