Commutation matrix

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.

Description

Consider the autonomous 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 $\mathrm{\Delta }t>0$:

$0={\tau }_0<{\tau }_1<\dots <{\tau }_N=T\text{ with }{\tau }_n:=n\mathrm{\Delta }t\text{ and }\mathrm{\Delta }t=\frac{T}{N};$

• set $Y_0=x_0;$

• recursively define $Y_n$ for $1\le n\le N$ by

$Y_{n+1}=Y_n+a(Y_n)\mathrm{\Delta }t+b(Y_n)\mathrm{\Delta }{W}_n+\frac{1}{2}b(Y_n)b^{\prime }(Y_n)((\mathrm{\Delta }{W}_n{)}^2-\mathrm{\Delta }t),$

where $b^{\prime }$ denotes the derivative of $b(x)$ with respect to $x$ and

$\mathrm{\Delta }{W}_n=W_{{\tau }_{n+1}}-W_{{\tau }_n}$

are independent and identically distributed normal random variables with expected value zero and variance $\mathrm{\Delta }t$. Then $Y_n$ will approximate $X_{{\tau }_n}$ for $0\le n\le N$, and increasing $N$ will yield a better approximation.

The error of the Milstein method is of order $\mathrm{\Delta }t$, which is considerably better than the Euler–Maruyama method, whose error is of order $(\mathrm{\Delta }t{)}^{1/2}$.

Intuitive derivation

For this derivation, we will only look at geometric Brownian motion (GBM), the stochastic differential equation of which is given by

$\mathrm{d}{X}_t=\mu X\mathrm{d}t+\sigma Xd{W}_t$

with real constants $\mu$ and $\sigma$. Using Itō's lemma we get

$\mathrm{d}\ln X_t=(\mu -\frac{1}{2}{\sigma }^2)\mathrm{d}t+\sigma \mathrm{d}{W}_t,$

Thus, the solution to the GBM SDE is

$\begin{array}{rl}{X}_{t+\mathrm{\Delta }t}& =X_t\exp \{\underset{t}{\overset{t+\mathrm{\Delta }t}{\int }}(\mu -\frac{1}{2}{\sigma }^2)\mathrm{d}t+\underset{t}{\overset{t+\mathrm{\Delta }t}{\int }}\sigma \mathrm{d}{W}_u\}\\ & \approx X_t(1+\mu \mathrm{\Delta }t-\frac{1}{2}{\sigma }^2\mathrm{\Delta }t+\sigma \mathrm{\Delta }{W}_t+\frac{1}{2}{\sigma }^2(\mathrm{\Delta }{W}_t{)}^2)\\ & =X_t+a(X_t)\mathrm{\Delta }t+b(X_t)\mathrm{\Delta }{W}_t+\frac{1}{2}b(X_t)b^{\prime }(X_t)((\mathrm{\Delta }{W}_t{)}^2-\mathrm{\Delta }t)\end{array}$

where

$a(x)=\mu x,b(x)=\sigma x$.

See also

• Euler–Maruyama method

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534