Existence of solution to SDE with perscribed initial and terminal conditions 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?  
 A: 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
A: Look at section 2.6 of this document is devoted to an answer to this question.  In there it is shown that given a diffusion process $Z_t$ with dynamics given by
$$
dZ_t = \mu_t(Z_t)dt + \sigma_t(Z_t)dW_t
$$
then there exists a process process $Y_t$ whose dynamics follow
$$
dY_t =  \left( 
\mu_t(Y_t) + \sigma_t^2(Y_t)\partial_xlog(p(y,T|Y_t,t)
\right)dt
+ \sigma_t(Y_t)dW_t
$$
where $p(y,T|x,t)$ is the conditional density of the process $Z_t$, 
satisfying $Y_0 = z_1$ and $Y_T=Z_2$, with the same volatility as $Z_t$, however it's drift is no longer given by $\mu_t$ alone.  

If however $\mu_t$ can be decomposed into the sum of two terms:
$$
\tilde{\mu}_t(x) + \sigma_t^2(Y_t)\partial_xlog(q(y,T|x,t)
$$
where
$$
dX_t = \tilde{\mu}_t(X_t)dt + \sigma_t(X_t)dW_t
$$
admits a strong solution and $q$ is it's conditional density then then things work.   
A: Here are some papers that are, I believe, relevant in this connection:


*

*http://dx.doi.org/10.1109/TAC.2015.2457784

*http://dx.doi.org/10.1109/TAC.2015.2457791

*http://arxiv.org/abs/1608.03622
