The SDEs \begin{equation}
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
\end{equation} with prescribed initial conditions are well studied. My question came up in my research and I have not found much on the subject. I was wondering under what conditions does there exist a solution to the above SDE on $[t_1,t_2]$ with the initial *and* terminal conditions:
$Z_{t_1}=z_1$ and $Z_{t_2}=z_2$?

Does there exist a good reference to these types of problems/ are these well studied? I know people use BSDEs usually to work with terminal conditions but when we have both initial and terminal conditions how can we approach the problem?

prescribedsingle point. On the other hand, you are always welcome to condition on the endpoint data (like that is done for the Brownian bridge), but I doubt that's what you had in mind. Am I missing something? $\endgroup$