Geometric Brownian motion

Remarks

By Itô's lemma, $S$ is the unique strong solution of $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$ with $S_0 = s_0$. Check: with $f(t, x) = s_0 \exp((\mu - \sigma^2/2) t + \sigma x)$, Itô's lemma gives $dS_t = (\mu - \sigma^2/2) S_t \, dt + \sigma S_t \, dW_t + \tfrac{1}{2} \sigma^2 S_t \, dt$, which simplifies to the equation above. The arithmetic process $s_0 + \mu t + \sigma W_t$ is not a geometric Brownian motion: it takes negative values with positive probability, while $S_t > 0$ always. For each $t$, $\log S_t$ is normal with mean $\log s_0 + (\mu - \sigma^2/2) t$ and variance $\sigma^2 t$, so $S_t$ is log-normal. Its mean is $E[S_t] = s_0 e^{\mu t}$, and for $\mu = 0$ the process is a martingale.