How can we show this process is an Ornstein-Uhlenbeck process?

Question

In my book it says that:

$X(t)=e^{-t}W(e^{2t})$

(where $t\in[0,\infty)$ and $W(t)$ is a Brownian motion) is an Ornstein-Uhlenbeck process. I know that $X(t)$ is a Gaussian process but how can we show that it is also an Ornstein-Uhlenbeck process?

Answer 1

Consider that $$X(t)-X(s)e^{-(t-s)}=e^{-t}(W(e^{2t})-W(e^{2s}))\sim\mathcal{N}(0,1-e^{-2(t-s)})$$ If $X(s)$ is known the conditional distribution becomes $$X(t)|X(s)\sim\mathcal{N}(X(s)e^{-(t-s)},1-e^{-2(t-s)})$$ If we indicate with $p(y,t|x,0)$ the Gaussian density of $X(t)|X(0)$, given the initial condition $X(0)=x\implies p(y,0|x,0)=\delta(y-x) $, it can be checked that $$\frac{\partial p}{\partial t}=\frac{\partial }{\partial y}(yp)+\frac{\partial^2 p}{\partial y^2}$$ which is the Fokker-Planck equation of the Ornstein-Uhlenbeck process with mean reversion coefficient $1$ and volatility $\sqrt{2}$.