Ornstein–Uhlenbeck process

For \(\alpha \in \mathbf R\) and \(\beta >0\),  Define \(X_t\) as the solution to the following SDE
\[dX_t = – \beta X_t dt + \alpha dW_t\]

where \(W_t\) is a standard Brownian Motion.

1.  Find \( d(e^{\beta t} X_t)\) using Ito’s Formula.
2. Use the calculation of   \( d(e^{\beta t} X_t)\) to show that
   \begin{align}  X_t = e^{-\beta t} X_0 + \alpha \int_0^t e^{-\beta(t-s)} dW_s\end{align}
3. Conclude that \(X_t\) is Gaussian process (see exercise: Gaussian Ito Integrals ). Find its mean and variance at time \(t\).
4. * Let \(h(t)\) and \(g(t)\) be  deterministic functions of time and let \(Y_t\) solve
   \[dY_t = – \beta Y_t dt + h(t)dt+ \alpha g(t) dW_t\]
   show find a formula analogous to part 2 above for \(Y_t\) and conclude that \(Y_t\) is still Gaussian. Find it mean and Variance.