Geometric Brownian Motion (GBM)ΒΆ
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 \(\varepsilon_{t}\sim N\left(0,1\right)\).