Euler-Maruyama method

In mathematics, the Euler-Maruyama method is a technique for the approximate numerical solution of a stochastic differential equation. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) A stochastic differential equation (SDE is a Differential equation in which one or more of the terms is a Stochastic process, thus resulting in a solution which is It is a simple generalization of the Euler method for ordinary differential equations to stochastic differential equations. In Mathematics and Computational science, the Euler method, named after Leonhard Euler, is a first order numerical procedure for solving In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its It is named after Leonhard Euler and Gisiro Maruyama. Gisiro Maruyama was a Japanese Mathematician, noted for his contributions to the study of Stochastic processes.

Consider the Itō stochastic differential equation

$\mathrm{d} X_t = a(X_t) \, \mathrm{d} t + b(X_t) \, \mathrm{d} W_t,$

with initial condition X₀ = x₀, where W_t stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time [0, T]. Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to Stochastic processes such as Brownian motion ( Wiener process) In Mathematics, in the field of Differential equations an initial value problem is an Ordinary differential equation together with specified value called In Mathematics, the Wiener process is a continuous-time Stochastic process named in honor of Norbert Wiener. Then the Euler-Maruyama approximation to the true solution X is the Markov chain Y defined as follows:

partition the interval [0, T] into N equal subintervals of width δ > 0:

$0 = \tau_{0} < \tau_{1} < \cdots < \tau_{N} = T \mbox{ and } \delta = T/N;$

set Y₀ = x₀;

recursively define Y_n for 1 ≤ n ≤ N by

$\, Y_{n + 1} = Y_{n} + a(Y_{n}) \delta + b(Y_{n}) \Delta W_{n},$

where

$\Delta W_{n} = W_{\tau_{n + 1}} - W_{\tau_{n}}.$

Note that the random variables ΔW_n are independent and identically distributed normal random variables with expected value zero and variance δ. In Mathematics, a Markov chain, named after Andrey Markov, is a Stochastic process with the Markov property. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way "IID" or "iid" redirects here For other uses see IID (disambiguation. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of

References

Kloeden, P. E. , & Platen, E. (1999). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. ISBN 978-3-540-54062-5.

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