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.

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