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 Wt 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:

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

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

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

Further reading