← Stochastic Differential Equations

Euler–Maruyama method

Euler–Maruyama is a basic numerical method for simulating an Itô SDE

\[dX_t=a(X_t,t)dt+b(X_t,t)dW_t.\]

For step size \(\Delta t\),

\[X_{n+1}=X_n+a(X_n,t_n)\Delta t+b(X_n,t_n)\Delta W_n,\]

where

\[\Delta W_n=\sqrt{\Delta t}\,Z_n,\qquad Z_n\sim N(0,1).\]

A new independent normal random number is generated for each step.

Key idea. Euler–Maruyama is Euler's method plus a Brownian increment representing the stochastic contribution during each time step.