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 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 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} < \cdots < \tau_{N} = T \mbox{ and } \delta = T/N;
  • set Y0 = x0;
  • recursively define Yn 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 ΔWn are independent and identically distributed normal random variables with expected value zero and variance δ.

[edit] 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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