Milstein method

In mathematics, the Milstein method, named after Grigori N. Milstein, 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 W_t 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:

$0 = \tau_0 < \tau_1 < \dots < \tau_N = T\text{ and }\delta = \frac{T}{N};$

• set Y₀ = x₀;

• recursively define Y_n 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 ΔW_n are independent and identically distributed normal random variables with expected value zero and variance δ.

See also

• Euler–Maruyama method

References

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