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Milstein method - Wikipedia

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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 Milstein who first published it in 1974.[1][2]

Consider the autonomous Itō stochastic differential equation: {\displaystyle \mathrm {d} X_{t}=a(X_{t})\,\mathrm {d} t+b(X_{t})\,\mathrm {d} W_{t}} with initial condition {\displaystyle X_{0}=x_{0}}, where {\displaystyle W_{t}} denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time {\displaystyle [0,T]}. Then the Milstein approximation to the true solution {\displaystyle X} is the Markov chain {\displaystyle Y} defined as follows:

Note that when {\displaystyle b'(Y_{n})=0} (i.e. the diffusion term does not depend on {\displaystyle X_{t}}) this method is equivalent to the Euler–Maruyama method.

The Milstein scheme has both weak and strong order of convergence {\displaystyle \Delta t} which is superior to the Euler–Maruyama method, which in turn has the same weak order of convergence {\displaystyle \Delta t} but inferior strong order of convergence {\displaystyle {\sqrt {\Delta t}}}.[3]

Intuitive derivation

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For this derivation, we will only look at geometric Brownian motion (GBM), the stochastic differential equation of which is given by: {\displaystyle \mathrm {d} X_{t}=\mu X\mathrm {d} t+\sigma XdW_{t}} with real constants {\displaystyle \mu } and {\displaystyle \sigma }. Using Itō's lemma we get: {\displaystyle \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: {\displaystyle {\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 {\displaystyle a(x)=\mu x,~b(x)=\sigma x}

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

Numerical solution for the stochastic differential equation where the drift is twice the diffusion coefficient.

Computer implementation

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The following Python code implements the Milstein method and uses it to solve the SDE describing geometric Brownian motion defined by {\displaystyle {\begin{cases}dY_{t}=\mu Y\,{\mathrm {d} }t+\sigma Y\,{\mathrm {d} }W_{t}\\Y_{0}=Y_{\text{init}}\end{cases}}} 

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

import numpy as np
import 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)
  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/
  • 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)