Milstein method

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.

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 $\Delta t>0$:

$0=\tau _{0}<\tau _{1}<\dots <\tau _{N}=T{\text{ with }}\tau _{n}:=n\Delta t{\text{ and }}\Delta t={\frac {T}{N}};$

• set $Y_{0}=x_{0};$

• recursively define $Y_{n}$ for $1\leq n\leq N$ by

$Y_{n+1}=Y_{n}+a(Y_{n})\Delta t+b(Y_{n})\Delta W_{n}+{\frac {1}{2}}b(Y_{n})b'(Y_{n})\left((\Delta W_{n})^{2}-\Delta t\right),$

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 $\Delta t$.

Intuitive Derivation

Using Ito's formula, we can rewrite the SDE at the top of the page as

$\mathrm {d} \ln X_{t}=\left(\mu (X_{t})-{\frac {1}{2}}\sigma (X_{t})^{2}\right)\mathrm {d} t+\sigma (X_{t})\mathrm {d} W_{t},$

where $\mu (x):=a(x)/x$ and $\sigma (x):=b(x)/x$. The solution to this SDE is

${\begin{aligned}X_{t+\Delta t}&=X_{t}\exp \left\{\int _{t}^{t+\Delta t}\left(\mu (X_{t})-{\frac {1}{2}}\sigma (X_{u})^{2}\right)\mathrm {d} t+\int _{t}^{t+\Delta t}\sigma (X_{u})\mathrm {d} W_{u}\right\}\\&\approx X_{t}\left(1+\mu (X_{t})\Delta t-{\frac {1}{2}}\sigma ^{2}(X_{t})\Delta t+\sigma (X_{t})\Delta W_{t}+{\frac {1}{2}}\sigma (X_{t})^{2}(\Delta W_{t})^{2}\right)\\&=X_{t}+a(X_{t})\Delta t+b(X_{t})\Delta W_{t}+{\frac {1}{2}}{\frac {b(X_{t})^{2}}{X_{t}}}((\Delta W_{t})^{2}-\Delta t)\\&\approx X_{t}+a(X_{t})\Delta t+b(X_{t})\Delta W_{t}+{\frac {1}{2}}b(X_{t})b'(X_{t})((\Delta W_{t})^{2}-\Delta t)\end{aligned}}$

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)