I was wondering if there has been some results for the following question:
Let $X(t)=\mu dt+\sigma dB_t$ ($t\ge0$, $X(0)=X_0$, $\sigma>0$, $\mu\ne 0$) be a diffusion process with constant drift and diffusion. Suppose $X_0\in (b,B)\subseteq \Re$. Define $\tau\equiv \inf\{t\,|\,X(t)\in \{b,B\}\}$ be the first-hitting time (an almost surely finite stopping time). Then what is the probability (density) that the diffusion process stops at time $t\ge 0$ at $b$ (namely, the density for the event $\{X(\tau)=b, \tau=t\}$)?
I hope someone can give me a hint or some references. Many thanks!