The Ornstein-Uhlenbeck Process: Stationary Distributions

Students frequently conflate the stationary variance with the time-dependent variance. Remember that while

\text{Var}(X_t)

converges to

\sigma^2 / 2\theta

t \to \infty

, the transient variance is strictly

\frac{\sigma^2}{2\theta}(1 - e^{-2\theta t})

, which only attains the stationary limit in the infinite time horizon.

Institutional Reference: Advanced Stochastic Processes