Stochastic Differential Equation for Time Integral of Stochastic Process

Question

Let $X(t)$ denote standard Brownian motion $dX(t) = a X dt + X dW(t)$ with solution $X(t) = e^{a t + W(t)}$. I want to consider the time-integrated process \begin{equation} Y(t) := \int_0^t d\tau~ X(\tau) \end{equation} which itself is again a stochastic process. My question is: How can I write a stochastic differential equation for $Y$ that is of the form \begin{equation} dY(t) = \ldots dt + \ldots dW(t) \end{equation} Whenever I use Ito's Lemma, I trivially obtain $dY(t) = X(t) dt$. Closely related to this problem: what would be the corresponding Fokker-Planck equation?