Euler–Maruyama method

Remarks

The increments $\Delta B_n$ are independent and normal with mean $0$ and variance $\Delta t$. In a simulation one takes $\Delta B_n = \sqrt{\Delta t} \, Z_n$ with independent standard normal $Z_n$. Example: for geometric Brownian motion, $a(t, x) = \mu x$ and $b(t, x) = \sigma x$, so $Y_{n+1} = Y_n (1 + \mu \Delta t + \sigma \Delta B_n)$. Non-example: the update $Y_{n+1} = Y_n + a(t_n, Y_n) \Delta t + b(t_n, Y_n) \Delta t \, Z_n$ uses noise of size $\Delta t$ instead of $\sqrt{\Delta t}$, so its noise vanishes in the limit. If $b = 0$, the scheme is the Euler method for the ordinary equation $x' = a(t, x)$. Assume that $a$ and $b$ are Lipschitz in $x$ uniformly in $t$, grow at most linearly, and are Hölder continuous of exponent $1/2$ in $t$. Then $E\abs{X_T - Y_N} \le C \sqrt{\Delta t}$ for a constant $C$.