Ornstein-Uhlenbeck process

$d{X}_t=\theta (\mu -x_t)dt+\sigma d{W}_t$, where

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

It satisfies the properties:

• $\mathbb{E}[S_t]=\mu -(\mu -S_0)e^{\theta t}$
• $\mathrm{Var}[S_t]=\frac{{\sigma }^2}{2\theta }\left(1-e^{-2\theta t}\right)$