Ito Variation of Constants

For  functions \(f(x)\) and\( g(x) \) and constant \(\beta>0\),  define \(X_t\) as the solution to the following SDE
\[dX_t = – \beta X_t dt + h(X_t)dt + g(X_t) dW_t\]

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

1.  Show that \(X_t\) can be written as
   \[X_t = e^{-\beta t} X_0 + \int_0^{t}  e^{-\beta (t-s)} h(X_s) ds +  \int_0^{t}  e^{-\beta (t-s)} g(X_s) dW_s\]
   See exercise:  Ornstein–Uhlenbeck process for guidance.
2. Assuming that \(|h(x)| < K\)  and \(|g(x)|<K\), show that  there exists a constant \(C(X_0)\) so that
   \[ \mathbf E [|X_t|] < C(X_0) \]
   for all \(t >0\). It might be convenient the remember the Cauchy–Schwarz inequality.
3. * Assuming that \(|h(x)| < K\)  and \(|g(x)|<K\), show that  for any integer \(p >0\) there exists a constant \(C(p,X_0)\) so that
   \[ \mathbf E [|X_t|^{2p}] < C(p,X_0) \]
   for all \(t >0\). See exercise: Ito Moments for guidance.