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.

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 $X_{0}=x_{0}$, 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 $b'$ denotes the derivative of $b(x)$ with respect to $x$ and:

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

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

Note that when $b'(Y_{n})=0$, i.e. the diffusion term does not depend on $X_{t}$ , this method is equivalent to the Euler–Maruyama method.

The Milstein scheme has both weak and strong order of convergence, $\Delta t$, which is superior to the Euler–Maruyama method, which in turn has the same weak order of convergence, $\Delta t$, but inferior strong order of convergence, ${\sqrt {\Delta t}}$.

Intuitive derivation

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

$\mathrm {d} X_{t}=\mu X\mathrm {d} t+\sigma XdW_{t}$

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

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

Thus, the solution to the GBM SDE is:

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

where

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

See numerical solution is presented above for three different trajectories.

Numerical solution for the stochastic differential equation just presented, the drift is twice the diffusion coefficient.

See also

• Euler–Maruyama method

Further reading

• 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)