Euler–Maruyama method

In Itô calculus, the Euler–Maruyama method (also called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations. It is named after Leonhard Euler and Gisiro Maruyama. Unfortunately, the same generalization cannot be done for any arbitrary deterministic method.

Consider the stochastic differential equation (see Itô calculus)

${\displaystyle \mathrm {d} X_{t}=a(X_{t},t)\,\mathrm {d} t+b(X_{t},t)\,\mathrm {d} W_{t},}$

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]. 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 ${\displaystyle \Delta t>0}$:

${\displaystyle 0=\tau _{0}<\tau _{1}<\cdots <\tau _{N}=T{\text{ and }}\Delta t=T/N;}$

• set Y₀ = x₀
• recursively define Y_n for 0 ≤ n ≤ N-1 by

${\displaystyle \,Y_{n+1}=Y_{n}+a(Y_{n},\tau _{n})\,\Delta t+b(Y_{n},\tau _{n})\,\Delta W_{n},}$

where

${\displaystyle \Delta W_{n}=W_{\tau _{n+1}}-W_{\tau _{n}}.}$

The random variables ΔW_n are independent and identically distributed normal random variables with expected value zero and variance ${\displaystyle \Delta t}$.