Milstein method

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In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation.

Consider the Itō stochastic differential equation

\mathrm{d} X_{t} = a(X_{t}) \, \mathrm{d} t + b(X_{t}) \, \mathrm{d} W_{t},

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 δ > 0:
0 = \tau_{0} < \tau_{1} < \dots < \tau_{N} = T and \delta = \frac{T}{N};
  • set Y0 = x0;
  • recursively define Yn for 1 \leq n \leq N by
Y_{n + 1} = Y_{n} + a(Y_{n}) \delta + b(Y_{n}) \Delta W_{n} + \frac{1}{2} b(Y_{n}) b'(Y_{n}) \left( (\Delta W_{n})^{2} - \delta \right),

where

\Delta W_{n} = W_{\tau_{n + 1}} - W_{\tau_{n}}

and b' denotes the derivative of b(x) with respect to x. Note that the random variables ΔWn are independent and identically distributed normal random variables with expected value zero and variance δ.

[edit] References

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