Denote by $B$ a Brownian bridge process, by $B(\omega)$ a realization of it and by $B_t$ the projection to the time point $t \in [0,1]$. Now let $c < 0$ and
$$t^*(\omega) = \sup\{t \in [0,1]: B_t(\omega) \leq c\}$$ (with $t^*(\omega)=0$ if the latter set is empty).
I am now interested in the distribution of $B_{t*}$, for example what is $\mathbb E[B_{t*}]$?
More generally, I am interested in references which explore such stopping times and mainly the expected value of the process evaluated at the stopping time, in cases where for example Doob's optional stopping theorem is no longer applicable.