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 (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. Unfortunately the same generalization cannot be done for the other methods from deterministic theory, e.g. Runge–Kutta schemes.

Consider the stochastic differential equation (see Itō calculus)

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:

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

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