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.

- Find \( d(e^{\beta t} X_t)\) using Ito’s Formula.
- 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} - Conclude that \(X_t\) is Gaussian process (see exercise: Gaussian Ito Integrals ). Find its mean and variance at time \(t\).
- * 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.