Milstein method

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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.[1][2]


Consider the autonomous Itō stochastic differential equation:

with initial condition , where stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time . Then the Milstein approximation to the true solution is the Markov chain defined as follows:

  • partition the interval into equal subintervals of width :
  • set
  • recursively define for by:

where denotes the derivative of with respect to and:

are independent and identically distributed normal random variables with expected value zero and variance . Then will approximate for , and increasing will yield a better approximation.

Note that when , i.e. the diffusion term does not depend on , this method is equivalent to the Euler–Maruyama method.

The Milstein scheme has both weak and strong order of convergence, , which is superior to the Euler–Maruyama method, which in turn has the same weak order of convergence, , but inferior strong order of convergence, .[3]

Intuitive derivation[edit]

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:


See numerical solution is presented above for three different trajectories.[4]

Numerical solution for the stochastic differential equation just presented, the drift is twice the diffusion coefficient.

See also[edit]


  1. ^ Mil'shtein, G. N. (1974). "Approximate integration of stochastic differential equations". Teoriya Veroyatnostei i ee Primeneniya (in Russian). 19 (3): 583–588.
  2. ^ Mil’shtein, G. N. (1975). "Approximate Integration of Stochastic Differential Equations". Theory of Probability & Its Applications. 19 (3): 557–000. doi:10.1137/1119062.
  3. ^ V. Mackevičius, Introduction to Stochastic Analysis, Wiley 2011
  4. ^ Umberto Picchini, SDE Toolbox: simulation and estimation of stochastic differential equations with Matlab.

Further reading[edit]

  • Kloeden, P.E., & Platen, E. (1999). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. ISBN 3-540-54062-8.CS1 maint: multiple names: authors list (link)