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]

Description[edit]

Consider the autonomous Itō stochastic differential equation

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 :
  • 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

where

.

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]

References[edit]

  1. ^ Mil'shtein, G. N. (1974). "Approximate integration of stochastic differential equations". Teor. Veroyatnost. i Primenen (in Russian). 19 (3): 583–588. 
  2. ^ Mil’shtejn, 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. http://sdetoolbox.sourceforge.net/

Further reading[edit]

  • Kloeden, P.E., & Platen, E. (1999). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. ISBN 3-540-54062-8.