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 ${\displaystyle b'(Y_{n})=0}$, 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, ${\displaystyle {\sqrt {\Delta t}}}$.^[3]

Intuitive derivation

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

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

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.^[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.