Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space $W = \{W_t:0<t<T\} $ be a Brownian Motion. Fix $\alpha,\beta \in \mathbb{R}$ and consider the following SDE:
$$dXt = \frac{\beta - X_t}{T-t}dt + dWt, \hspace{5mm} 0<t<T$$ $$X_0 = \alpha, \hspace{5mm} X_T = \beta.$$
Apply Ito's Lemma to $f(t,X_t) = \frac{X_t}{T-t}$ and find the distribution of $X_t$.
I got the solution to be $$X_t = \frac{X_0(T-t)}{T} + \frac{\beta t}{T} + (T-t) \int_{0}^{t}\frac{1}{T-s} dWs$$ Which is normally distributed $$X_t \thicksim N(\frac{\alpha(T-t)}{T}+ \frac{\beta t}{T}, \frac{(T-t)t}{T})$$
Based off of bits of info I've found online, I'm unsure if my Expected Value/Solution is correct.