# 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?

• 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

## 3 Answers

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:

• Thank you very much. I was wondering, is there a reference which deals with these constructions for diffusions driven by cylindrical Wiener processes? This is the particular version I would need, if it exists. – AIM_BLB Aug 15 '16 at 19:19
• The Wiener processes in the SPDEs appearing in those references are all cylindrical. – Nawaf Bou-Rabee Aug 15 '16 at 19:28

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.

Here are some papers that are, I believe, relevant in this connection: