Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). The solution of the \(\mathrm{SDE}\)

\[d X_{t}=-\beta X_{t} d t+\sigma d B_{t}, \quad X_{0}=\xi\]

with \(\beta>0\) and \(\sigma \in \mathbb{R}\) is an Ornstein-Uhlenbeck process.

a) Find \(X_{t}\) explicitly.

b) Determine the distribution of \(X_{t}\). Is \(\left(X_{t}ight)_{t \geqslant 0}\) a Gaussian process? Find the correlation function \(C(s, t)=\mathbb{E} X_{S} X_{t}, s, t \geqslant 0\).

c) Determine the asymptotic distribution \(\mu\) of \(X_{t}\) as \(t ightarrow \infty\).

d) Assume that the initial condition \(\xi\) is a random variable which is independent of \(\left(B_{t}ight)_{t \geqslant 0}\), and assume that \(\xi \sim \mu\) with \(\mu\) from above. Find the characteristic function of \(\left(X_{t_{1}}, \ldots, X_{t_{n}}ight)\) for all \(0 \leqslant t_{1}

e) Show that the process \(\left(U_{t}ight)_{t \geqslant 0}\), where \(U_{t}:=\frac{\sigma}{\sqrt{2 \beta}} e^{-\beta t} B\left(e^{2 \beta t}-1ight)\), has the same finite dimensional distributions as \(\left(X_{t}ight)_{t \geqslant 0}\) with initial condition \(X_{0}=0\). Compare this with the stationary OU process from Problem 2.11.