Stationary OU process

Question

Let $$ Y_t = \frac{\sigma}{\sqrt{\mu}} e^{-\mu t}W(e^{2\mu t}), $$ where $W(t)\equiv W_t$ is the standard Brownian motion, $\sigma,\mu>0$. I wonder how to show that $Y_t$ is a stationary Ornstein-Uhlenbeck process. I think maybe I can show that it's a stationary Gaussian process with $\mathbb{E}[Y_{\tau+s}Y_s]=\mathbb{E}[Y_{\tau}Y_0]$.

Answer 1

Hint : Go the other way round and show that if $Y(t)$ is an Ornstein-Uhlenbeck process then the process $Z_t$ given by

\begin{align} Z_t =t^{\frac12}\frac{\sqrt{2\mu}}{\sigma} Y\left(\frac{1}{2\mu}\log t\right) \, , \end{align} is a standard Brownian motion.