# Existence of solution to SDE with perscribed initial and terminal conditions

The SDEs $$dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t$$ 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?

• So, what exactly do you mean by that? If you prescribe the initial condition, then, provided that the functions $\mu$ and $\sigma$ are decent enough, you get a unique stochastic process satisfying the equation and its distribution at $t_2$ is almost never concentrated at a single point, much less often at a prescribed single 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? – fedja Aug 15 '16 at 18:30
• I had something like a Brownian bridge in mind. I was wondering if the same idea existed for any diffusion process? – AIM_BLB Aug 15 '16 at 18:54
• you might find some interesting results if you search for "Schrodinger Bridge" – Piyush Grover Aug 15 '16 at 19:52

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.