Milstein method

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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[edit]

Consider the autonomous Itō stochastic differential equation:

with initial condition , where stands for 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:
    where denotes the derivative of with respect to and:
    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[edit]

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

with real constants and . Using Itō's lemma we get:

Thus, the solution to the GBM SDE is:

where

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.

Computer implementation[edit]

The following Python code implements the Milstein method and uses it to solve the SDE describing the Geometric Brownian Motion defined by

# -*- 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[0] = y_init

# Loop
for _ in range(num_sims):
    for i in range(1, ts.size):
        t = (i - 1) * dt
        y = ys[i - 1]
        # Milstein method
        dw_ = dW(dt)
        ys[i] = 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[edit]

References[edit]

  1. ^ Mil'shtein, G. N. (1974). "Approximate integration of stochastic differential equations". Teoriya Veroyatnostei i ee Primeneniya (in Russian). 19 (3): 583–588.
  2. ^ Mil’shtein, 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[edit]

  • Kloeden, P.E., & Platen, E. (1999). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. ISBN 3-540-54062-8.{{cite book}}: CS1 maint: multiple names: authors list (link)