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 it in 1974.

Description

Consider the autonomous Itō stochastic differential equation:
$\mathrm X_t = a(X_t) \, \mathrm t + b(X_t) \, \mathrm W_t$
with initial condition $X_ = x_$, where $W_$ stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time  ,T/math>. Then the Milstein approximation to the true solution $X$ is the Markov chain $Y$ defined as follows:

* partition the interval ,T/math> into $N$ equal subintervals of width $\Delta t>0$: $0 = \tau_0 < \tau_1 < \dots < \tau_N = T\text\tau_n:=n\Delta t\text\Delta t = \frac$
* set $Y_0 = x_0;$
* recursively define $Y_n$ for $1 \leq n \leq N$ by: $Y_ = Y_n + a(Y_n) \Delta t + b(Y_n) \Delta W_n + \frac b(Y_n) b'(Y_n) \left( (\Delta W_n)^2 - \Delta t \right)$ where $b'$ denotes the derivative of $b(x)$ with respect to $x$ and: $\Delta W_n = W_ - W_$ are independent and identically distributed normal random variables with expected value zero and variance $\Delta t$. Then $Y_n$ will approximate $X_$ 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 $X_$, 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, $\sqrt$.

Intuitive derivation

For this derivation, we will only look at geometric Brownian motion (GBM), the stochastic differential equation of which is given by:
$\mathrm X_t = \mu X \mathrm t + \sigma X d W_t$
with real constants $\mu$ and $\sigma$. Using Itō's lemma we get:
$\mathrm\ln X_t= \left(\mu - \frac \sigma^2\right)\mathrmt+\sigma\mathrmW_t$

Thus, the solution to the GBM SDE is:
$\begin X_&=X_t\exp\left\ \\ &\approx X_t\left(1+\mu\Delta t-\frac \sigma^2\Delta t+\sigma\Delta W_t+\frac\sigma^2(\Delta W_t)^2\right) \\ &= X_t + a(X_t)\Delta t+b(X_t)\Delta W_t+\fracb(X_t)b'(X_t)((\Delta W_t)^2-\Delta t) \end$
where
$a(x) = \mu x, ~b(x) = \sigma x$

See numerical solution is presented above for three different trajectories.Umberto Picchini, SDE Toolbox: simulation and estimation of stochastic differential equations with Matlab. http://sdetoolbox.sourceforge.net/

Computer implementation

The following Python code implements the Milstein method and uses it to solve the SDE describing the Geometric Brownian Motion defined by
$\begin dY_t= \mu Y \, t + \sigma Y \, W_t \\ Y_0=Y_\text \end$

# -*- coding: utf-8 -*-
# Milstein Method

import numpy as np
import matplotlib.pyplot as plt

num_sims = 1 # One Example

# One Second and thousand grid points
t_init, t_end = 0, 1
N = 1000 # Compute 1000 grid points
dt = float(t_end - t_init) / N

## Initial Conditions
y_init = 1
μ, σ = 3, 1

# dw Random process
def dW(Δt):
"""Random sample normal distribution"""
return np.random.normal(loc=0.0, scale=np.sqrt(Δt))

# vectors to fill
ts = np.arange(t_init, t_end + dt, dt)
ys = np.zeros(N + 1)
ys = y_init

# Loop
for _ in range(num_sims):
for i in range(1, ts.size):
t = (i - 1) * dt
y = ys - 1 # Milstein method
dw_ = dW(dt)
ys = y + μ * dt * y + σ * y * dw_ + 0.5 * σ**2 * y * (dw_**2 - dt)
plt.plot(ts, ys)

# Plot
plt.xlabel("time (s)")
plt.grid()
h = plt.ylabel("y")
h.set_rotation(0)
plt.show()

See also

* Euler–Maruyama method

References

Further reading

* {{cite book , author=Kloeden, P.E., & Platen, E. , title=Numerical Solution of Stochastic Differential Equations , publisher=Springer, Berlin , year=1999 , isbn=3-540-54062-8

Numerical differential equations
Stochastic differential equations