Euler-Maruyama method

In mathematics, the Euler-Maruyama method is a technique for the approximate numerical solution of a stochastic differential equation. 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 Itō stochastic differential equation

:$mathrm\{d\}\; X\_t\; =\; a(X\_t)\; ,\; mathrm\{d\}\; t\; +\; b(X\_t)\; ,\; mathrm\{d\}\; W\_t,$

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:

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

::

* set "Y"₀ = "x"₀;

* 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

::$Delta\; W\_\{n\}\; =\; W\_\{\; au\_\{n\; +\; 1\; -\; W\_\{\; au\_\{n.$

Note that the random variables Δ"W"_"n" are independent and identically distributed normal random variables with expected value zero and variance "δ".

References

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