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 $Y_{0}=x_{0};$

• 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 $\Delta W_{n}$ are independent and identically distributed normal random variables with expected value zero and variance $\delta$.

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.{{cite book}}: CS1 maint: multiple names: authors list (link)