Geometric Brownian Motion (GBM)
Suppose that \(S_{t}\) evolves according to
\[
\frac{dS_{t}}{S_{t}}=\mu dt+\sigma dW_{t}.
\]
In logs:
\[
d\log S_{t}=\left(\mu-\frac{1}{2}\sigma^{2}\right)dt+\sigma dW_{t}.
\]
After integration on the interval \(\left[t,t+h\right]\):
\[
r_{t,h}=\log\frac{S_{t+h}}{S_{t}}
=\left(\mu-\frac{1}{2}\sigma^{2}\right)h
+\sigma\sqrt{h}\varepsilon_{t+h},
\]
where \(\varepsilon_{t}\sim N\left(0,1\right)\).