Euler-Maruyama method

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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 G. Maruyama.

Consider the Itō stochastic differential equation

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

with initial condition X0 = x0, where Wt 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:
0 = \tau_{0} < \tau_{1} < \dots < \tau_{N} = T and \delta = \frac{T}{N};
  • set Y0 = x0;
  • recursively define Yn for 1 \leq n \leq 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 ΔWn are independent and identically distributed normal random variables with expected value zero and variance δ.

[edit] Reference

  • Kloeden, P.E., & Platen, E. (1999). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. ISBN 3-540-54062-8. 
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