A stochastic process {X(t), 0} is said to be stationary if X(t), ... X() has the same joint distribution as X(t, + a),., X(1, +

a) for all n,

a, t, t

(a) Prove that a necessary and sufficient condition for a Gaussian pro- cess to be stationary is that Cov(X(s), X(t)) depends only on t-s st, and E[X(t)] =

c.

(b) Let {X(t), 0} be Brownian motion and define -ut/2 V(t) = X(ae") Show that {V(t), 10} is a stationary Gaussian process It is called the Ornstein-Uhlenbeck process.