How do you construct an Ito Differential Equation with solution $Y_p$, where $p \in \mathbb{Z}+$ and $Y_p = (X(t) - \mathbb{E}(X(t)))^p$
Where $X(t)$ is the Ornstein- Uhlenbeck process, i.e
$dX = - \gamma X dt + \sigma dW$
and
$X(t) = \exp(-\gamma t)X(0) + \sigma \int_{0}^{t} \exp (-\gamma (t-s)) dW(s)$
I know that
$\mathbb{E}(X(t)) = X(0)\exp(-\gamma t)$
and so
$\lim\limits_{t \to \infty} \mathbb{E}(X(t)) = 0$
I'm just not sure where to go from here.
Let $Z(t) := \sigma \int_{0}^{t} \exp (-\gamma (t-s)) dW(s)$. We have $Y_p(t) = Z(t)^p$ and we want to find the SDE solved by $Y_p$.
Clearly, $Z(t)$ is a driftless OU-process, i.e. it is a solution of the SDE $$dZ(t) = - \gamma Z(t) dt + \sigma dW(t),\quad Z(0) = 0 $$
Itô's lemma applied to $Y_p(t) = f(t,Z(t))$ with $f(t,x) := x^p$ yields \begin{align} dY_p(t) =df(t,Z(t)) &= 0\times dt + \frac{\partial f}{\partial x}(t,Z_t)dZ(t) + \frac 1 2\frac{\partial^2 f}{\partial x^2}(t,Z_t)dt\\ &=pZ(t)^{p-1}dZ(t) + \frac{p(p-1)}{2}Z(t)^{p-2} dt\\ &=\left[-\gamma p Z(t)^p+ \frac{p(p-1)}{2}Z(t)^{p-2}\right]dt + \sigma p Z(t)^{p-1}dW(t) \end{align}
Finally, if we recall that $Z(t) = Y_p(t)^{1/p}$, we find that $Y_p(t)$ is solution of $$dY_p(t) = \left[-\gamma p Y_p(t)+ \frac{p(p-1)}{2}Y_p(t)^{(p-2)/p}\right]dt + \sigma p Y_p(t)^{(p-1)/p}dW(t)$$