Milstein method explained

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.^{[1] [2]}

Description

, where denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time  . Then the Milstein approximation to the true solution is the Markov chain defined as follows:

• Partition the interval

into equal subintervals of width :

• Set

• Recursively define

for by: $$Y\_\; =\; Y\_n\; +\; a(Y\_n)\; \backslash Delta\; t\; +\; b(Y\_n)\; \backslash Delta\; W\_n\; +\; \backslash frac\; b(Y\_n)\; b\text{'}(Y\_n)\; \backslash left((\backslash Delta\; W\_n)^2\; -\; \backslash Delta\; t\; \backslash right)$$ where denotes the derivative of with respect to and: $$\backslash Delta\; W\_n\; =\; W\_\; -\; W\_$$ are independent and identically distributed normal random variables with expected value zero and variance . Then will approximate for , and increasing will yield a better approximation.

Note that when

(i.e. the diffusion term does not depend on ) this method is equivalent to the Euler–Maruyama method.

The Milstein scheme has both weak and strong order of convergence

which is superior to the Euler–Maruyama method, which in turn has the same weak order of convergence but inferior strong order of convergence .^[3]

Intuitive derivation

For this derivation, we will only look at geometric Brownian motion (GBM), the stochastic differential equation of which is given by:$$\backslash mathrm\; X\_t\; =\; \backslash mu\; X\; \backslash mathrm\; t\; +\; \backslash sigma\; X\; d\; W\_t$$with real constants

and . Using Itō's lemma we get:$$\backslash mathrm\backslash ln\; X\_t=\; \backslash left(\backslash mu\; -\; \backslash frac\; \backslash sigma^2\backslash right)\backslash mathrmt+\backslash sigma\backslash mathrmW\_t$$

Thus, the solution to the GBM SDE is:where$$a(x)\; =\; \backslash mu\; x,\; ~b(x)\; =\; \backslash sigma\; x$$

The numerical solution is presented in the graphic for three different trajectories.^[4]

Computer implementation

The following Python code implements the Milstein method and uses it to solve the SDE describing geometric Brownian motion defined by$$\backslash begindY\_t=\; \backslash mu\; Y\; \backslash ,\; t\; +\; \backslash sigma\; Y\; \backslash ,\; W\_t\; \backslash \backslash Y\_0=Y\_\backslash text\backslash end$$

1. -*- coding: utf-8 -*-
2. Milstein Method

import numpy as npimport matplotlib.pyplot as plt

class Model: """Stochastic model constants.""" mu = 3 sigma = 1

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

def run_simulation: """ Return the result of one full simulation.""" # One second and thousand grid points T_INIT = 0 T_END = 1 N = 1000 # Compute 1000 grid points DT = float(T_END - T_INIT) / N TS = np.arange(T_INIT, T_END + DT, DT)

Y_INIT = 1

# Vectors to fill ys = np.zeros(N + 1) ys[0] = Y_INIT for i in range(1, TS.size): t = (i - 1) * DT y = ys[i - 1] dw = dW(DT)

# Sum up terms as in the Milstein method ys[i] = y + \ Model.mu * y * DT + \ Model.sigma * y * dw + \ (Model.sigma**2 / 2) * y * (dw**2 - DT)

return TS, ys

def plot_simulations(num_sims: int): """Plot several simulations in one image.""" for _ in range(num_sims): plt.plot(*run_simulation)

plt.xlabel("time (s)") plt.ylabel("y") plt.grid plt.show

if __name__

"__main__": NUM_SIMS = 2 plot_simulations(NUM_SIMS)

See also

• Euler–Maruyama method

Further reading

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

Notes and References

1. G. N.. Mil'shtein . Approximate integration of stochastic differential equations. Teoriya Veroyatnostei i ee Primeneniya. 19. 3. 1974. 583–588. ru.
2. Mil’shtein . G. N. . Approximate Integration of Stochastic Differential Equations . 10.1137/1119062 . Theory of Probability & Its Applications . 19 . 3 . 557–000 . 1975 .
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/