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.^[1][2]

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 $\Delta t>0$:

$0 = \tau_0 < \tau_1 < \dots < \tau_N = T\text{ with }\tau_n:=n\Delta t\text{ and }\Delta t = \frac{T}{N};$

• 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, that has the same weak order of convergence $\Delta t$, but inferior strong order of convergence $\sqrt{\Delta t}$.^[3]

Intuitive derivation

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

$\mathrm{d} X_t = \mu X \mathrm{d} t + \sigma X d W_t$

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

$\begin{align} X_{t+\Delta t}&=X_t\exp\left\{\int_t^{t+\Delta t}\left(\mu-\frac{1}{2}\sigma^2\right)\mathrm{d}t+\int_t^{t+\Delta t}\sigma\mathrm{d}W_u\right\} \\ &\approx X_t\left(1+\mu\Delta t-\frac{1}{2}\sigma^2\Delta t+\sigma\Delta W_t+\frac{1}{2}\sigma^2(\Delta W_t)^2\right) \\ &= X_t + a(X_t)\Delta t+b(X_t)\Delta W_t+\frac{1}{2}b(X_t)b'(X_t)((\Delta W_t)^2-\Delta t) \end{align}$

where

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

See numerical solution is presented above for three different trajectories.^[4]

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

See also

• Euler–Maruyama method

References

1. Mil'shtein, G. N. (1974). "Approximate integration of stochastic differential equations". Teor. Veroyatnost. i Primenen (in Russian) 19 (3): 583–588.
2. Mil’shtejn, G. N. (1975). "Approximate Integration of Stochastic Differential Equations". Theory of Probability & Its Applications 19 (3): 557–000. doi:10.1137/1119062.
3. V. Mackevičius, Introduction to Stochastic Analysis, Wiley 2011
4. Umberto Picchini, SDE Toolbox: simulation and estimation of stochastic differential equations with Matlab. http://sdetoolbox.sourceforge.net/

Further reading

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