To begin with, I have the the SDE for a 3D Bessel bridge between $(x,t) = (0,0)$ and $(x,t)=(0,T)$, which is given by $$dX_t = \left(\frac{1}{X_t}-\frac{X_t}{T-t}\right)dt + dB_t$$ where $B_t$ is a Brownian motion.
Now, if I have the following SDE (with $a\neq 1$): $$dX_t = \left(\frac{1}{X_t}-\frac{aX_t}{T-t}\right)dt + dB_t,$$ is this still a 3D Bessel bridge connecting $(x,t) = (0,0)$ and $(x,t)=(0,T)$? How can I map the later SDE to the former?
By the way, what is the general SDE of a 3D Bessel bridge connecting $(x,t)=(X_0,0)$ and $(x,t)=(X_T,T)$, with predefined $X_0$ and $X_T$? (I tried to google it, but got nothing)