Ornstein–Uhlenbeck process

Remarks

The explicit formula solves the equation: Itô's lemma applied to $e^{\theta t} X_t$ gives $d(e^{\theta t} X_t) = \theta \mu e^{\theta t} \, dt + \sigma e^{\theta t} \, dW_t$, which integrates to the formula. If $X_0$ is constant, $X_t$ is normal with mean $\mu + (X_0 - \mu) e^{-\theta t}$ and variance $\frac{\sigma^2}{2\theta}(1 - e^{-2\theta t})$. If $X_0$ is normal with mean $\mu$ and variance $\sigma^2 / (2\theta)$, the process is stationary, with covariance $\frac{\sigma^2}{2\theta} e^{-\theta \abs{t - s}}$. For $\theta = 0$, which the definition excludes, the equation gives $X_0 + \sigma W_t$, a Brownian motion with no mean reversion and no stationary law. Up to linear changes of space and time variables, it is the only non-trivial process that is stationary, Gaussian and Markov.