Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck (OU) process is defined by the SDE: $$dX_t = \theta(\mu - X_t)\, dt + \sigma\, dW_t$$ where $\theta > 0$ is the speed of mean reversion, $\mu$ is the long-run mean, and $\sigma$ is the volatility. Intuition: When $X_t > \mu$, the drift $\theta(\mu - X_t)$ is negative, pushing the process down toward $\mu$. When $X_t < \mu$, the drift is positive, pulling it back up. The process "orbits" around $\mu$ with random noise $\sigma\, dW_t$. Explicit solution: Starting from $X_0 = x_0$: $$X_t = \mu + (x_0 - \mu)e^{-\theta t} + \sigma \int_0^t e^{-\theta(t-s)}\, dW_s$$ The process is Gaussian with: $$E[X_t] = \mu + (x_0 - \mu)e^{-\theta t}, \quad \text{Var}(X_t) = \frac{\sigma^2}{2\theta}(1 - e^{-2\theta t})$$ Stationary distribution: As $t \to \infty$, $X_t \to N\!\left(\mu,\, \frac{\sigma^2}{2\theta}\right)$. Higher mean-reversion speed $\theta$ means lower stationary variance — the process stays closer to $\mu$. Concrete example: An interest rate model with $\mu = 5\%$, $\theta = 0.5$, $\sigma = 1\%$. Starting at 7%, the expected rate after 2 years is $5 + 2e^{-1} \approx 5.74\%$, reverting toward 5%. When to use: Modeling interest rates (Vasicek model), pairs trading (spread between two cointegrated stocks), and any quantity that reverts to a long-run level. The OU process is the continuous-time analog of the discrete AR(1) process.