[[Stationary Process|Stationary]] [[Stochastic Processes|stochastic process]] that is a Gaussian process, a Markov process and temporally homogeneous. >[!info] Definition - Ornstein-Uhlenbeck Process Process of the form$dX_t=\Theta(\mu-X_t)dt+\sigma dW_t,$with ... >- $\Theta \in \mathbb{R}_+$ as the mean-reversion rate >- $\sigma \in \mathbb{R}_+$ as the volatility >- $\mu \in \mathbb{R}$ as the long term mean The [[Stationary Process|stationarity]] is a result of the bounded variance. --- The [[Euler-Maruyama Method|Euler-Maruyama]] discretization is given via$X_{n+1}=X_n+\Theta(\mu-X_n)\Delta t+\sigma \sqrt{\Delta t}W_n,\quad W_n \sim \mathcal{N}(0,1)$