Let $X_t$ be an Ito diffusion process with initial condition $X_0 = x_0$. Let $T>0$ we a fixed deterministic time, and consider for 0 \leqslant t < T the process $Y_t = X_t| X_T = x_T$. Is this process an Ito process ? If so, how can one find its SDE ?
For Wiener process the answer is in affirmative: $ \mathrm{d} Y_t = -\frac{Y_t}{T-t}\mathrm{d} t + \sigma \mathrm{d} W_t , \quad Y_0 = 0 $ I suspect it might be the case in general (i.e. Ito bridge is an Ito process), at least for time-homogeneous Ito diffusions, using stochastic changes of time. I am hoping for a reference to a relevant books/articles.
Thank you.