Are Ito bridges themselves Ito processes?

Let $X_t$ be an Ito diffusion process with initial condition $X_0 = x_0$. Let $T>0$ we a fixed deterministic time, and consider for $0 \leqslant t < T$ the process $Y_t = X_t| X_T = x_T$. Is this process an Ito process ? If so, how can one find its SDE ?

For Wiener process the answer is in affirmative: $$\mathrm{d} Y_t = -\frac{Y_t}{T-t}\mathrm{d} t + \sigma \mathrm{d} W_t , \quad Y_0 = 0$$ I suspect it might be the case in general (i.e. Ito bridge is an Ito process), at least for time-homogeneous Ito diffusions, using stochastic changes of time. I am hoping for a reference to a relevant books/articles.

Thank you.

Under mild assumptions, a conditioned diffusion is still a diffusion. Indeed, the diffusion $dX = \mu(X) \, dt + \sigma(X) \, dW$ conditioned on the event $X_T=y_T$ follows the stochastic differential equation $$dX = \Big( \mu(X) + \sigma^2(X) \, \frac{\partial_x h(t,x,y_T)}{h(t,x,y_T)} \Big)\, dt + \sigma(X) \, dW$$ where $h(t,x,y) \, dy = \mathbb{P}(X_T \in dy \,| X_t=x)$ is the density of $X_T$ conditioned on the event $X_t = x$. This is a Doob h-transform. More can be read here.

You can have a look at Section 1.1 of the following paper and for the case you are interested in (under some conditions) I think that Example 8 is giving a satisfying (theoretical) answer to your question.

Best regards

• @ Sasha : you can also try to look at this article : proba.jussieu.fr/mathdoc/textes/PMA-649.pdf, best regards. Commented Apr 12, 2012 at 19:41
• Both links are broke. Could you please update them? Thank you.
– Hans
Commented Jul 27, 2017 at 7:30
• try here : proba.jussieu.fr/mathdoc/preprints/… Commented Jul 28, 2017 at 8:29
• The link does not lead to the paper. This citeseerx.ist.psu.edu/viewdoc/… does.
– Hans
Commented Jul 28, 2017 at 9:00
• google it you'll find it Commented Jul 28, 2017 at 12:27