I am curious about and have been trying for a long time the following question. I was wondering if there will be any hints or references that could help.
Assume that $B_t$ represents a standard Brownian motion and consider an adapted diffusion process $\{X_t\}_{t\ge 0}$, which solves the S.D.E. $$d X_t=\sigma X_t(1-X_t)d B_t$$ for some $\sigma>0$. Assume that $X_0=x\in (a,b)\subseteq (0,1)$. Suppose the two bounds $a$ and $b$ are absorbing, that is, the process is terminated once $X_t$ hits either $a$ or $b$. We can show that this defines a stopping time that is almost surely finite.
Let $T_a$ be the time at which $X_t$ reaches $a$ and $T_b$ the time at which $X_t$ reaches $b$. Let $\delta\in (0,1)$. Then is there a way to precisely compute, say, $\mathbf E[\delta^{T_a}]$?
Any hints or references will be appreciated!