# Okishio's theorem

In mathematics, the **Euler–Maruyama method** is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is a simple generalization of the Euler method for ordinary differential equations to stochastic differential equations. It is named after Leonhard Euler and Gisiro Maruyama.

Consider the stochastic differential equation (see Itō calculus)

with initial condition *X*_{0} = *x*_{0}, 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 **Euler–Maruyama approximation** to the true solution *X* is the Markov chain *Y* defined as follows:

- set
*Y*_{0}=*x*_{0};

- recursively define
*Y*_{n}for 1 ≤*n*≤*N*by

- where

The random variables Δ*W*_{n} are independent and identically distributed normal random variables with expected value zero and variance .

## Example

The following Python code implements Euler–Maruyama to solve the Ornstein–Uhlenbeck process:

import numpy as np import matplotlib.pyplot as plt import math import random tBegin=0 tEnd=2 dt=.00001 t = np.arange(tBegin, tEnd, dt) N = t.size IC=0 theta=1 mu=1.2 sigma=0.3 sqrtdt = math.sqrt(dt) y = np.zeros(N) y[0] = IC for i in xrange(1,N): y[i] = y[i-1] + dt*(theta*(mu-y[i-1])) + sigma*sqrtdt*random.gauss(0,1) ax = plt.subplot(111) ax.plot(t,y) plt.show()

## See also

## References

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