I found the following statement but can't figure out how to show it: Let
Y_t = -k Y_t dt + a dt + dW_t
be an Ornstein Uhlenbeck process, where k > 0, a \in R, W is Brownian motion.
Then it is an ergodic Gauss-Markov process, so there exists a \beta > 0 such that
sup_{t \geq 0} E(e^{\beta Y_t^2}) < \infty.
I can't explain why this should be true. can someone help me?
I tried to calculate
E(e^{\beta X_t^2})
for some Gaussian variable X_t directly, but this is not a finite value...