# 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.

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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.