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? 



Existence of solution to SDE with perscribed initial and terminal conditions
The keywords are **diffusion bridges** or **conditioned diffusions**. I like to think of the distribution of these bridges as the stationary distribution of an SPDE on the path space of the diffusion. For an intro to this viewpoint and practical methods for sampling from this distribution, check out the following references:


* M.G. Reznikoff, E. Vanden-Eijnden. "Invariant measures of stochastic partial differential equations and conditioned diffusions." C. R. Acad. Sci. Paris, Ser. I 340 (2005).
<http://www.math1.rwth-aachen.de/~westd/crass.pdf>
* Alexandros Beskos, Gareth O. Roberts, Andrew M. Stuart and Jochen Voss. "MCMC Methods for Diffusion Bridges." Stochastics and Dynamics, vol. 8, no. 3, pp. 319–350, 2008.
<https://homepages.warwick.ac.uk/~masdr/JOURNALPUBS/stuart74.pdf>
* Hairer, Martin, Andrew M. Stuart, and Jochen Voss. "Sampling conditioned diffusions." Trends in stochastic analysis 353 (2009): 159-186. 
<http://www.hairer.org/papers/HSV07.pdf>


