Geometric Brownian Motion

$d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, where

• $S_t$ is a Semimartingale
• $\mu ,\sigma \in \mathbb{R}$
• $W_t$ is a Wiener Process

It satisfies the properties:

• $S_t>0\forall t{\mathbb{R}}_0^{+}$
• $\mathbb{E}[S_t]=S_0{e}^{\mu t}$
• $\mathrm{Var}[S_t]=S_0^2{e}^{2\mu t}\left(e^{{\sigma }^{2}t}-1\right)$