I am curious about the following problem:
Let the diffusion process $\{X_t\}_{t\ge 0}$ be defined as $$dX_t=c(1-X_t)X_td B_t$$ where $X_0\in (a,b)\subseteq (0,1)$, $c>0$, and $B_t$ is the standard Brownian motion. Define the (first) hitting time $\tau\equiv\inf\{t\ge 0\,|\,X_t\notin(a,b)\}$.
My question is: How to derive the conditional densities $$\frac{\mathrm d\,\mathbb P(\tau\le t\,|\, X_\tau=b)}{\mathrm d t}\text{ and }\,\,\frac{\mathrm d\,\mathbb P(\tau\le t\,|\, X_\tau=a)}{\mathrm d t}?$$ Can anyone give me some hint, suggestion, or reference for this? Many thanks!
My current situation: (should be ignored if it is useless...) We can show that the process $$Y_t=-\frac{t}{2}+\ln\frac{X_t}{1-X_t}$$ is a standard Brownian motion under the changed measure $\mathbb Q$ with the Radon-Nikodym derivative $\mathrm d\mathbb Q/\mathrm d\mathbb P=X_t/X_0$.