Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 219 / 2 / b

For $t_1<t_2<t_3$, the condition from part a becomesIt holds for arbitrary positive time gaps exactly when $\eta =1$. The resulting exponential covariance function is the covariance of a stationary Ornstein-Uhlenbeck process, hence has the Markov property.

Writing $a_{32}=e^{-(t_3-t_2)/\tau }$, the predictive law isAs $t_3\to \infty$, $a_{32}\to 0$, so the predictive mean tends to the stationary mean $\mu$ and the predictive variance tends to the stationary variance $1$.