Consider a geometric Brownian motion $$ \frac{dA_t}{A_t} =\mu \, dt + \sigma \, dz_t$$ with $z_t$: Wiener process, $t\in [0,T]$.
Define a hitting time
$$ \tau_{uc} = \inf\{t\in [0,T]: A_t \leq A_{uc} \} $$
for some $A_{uc}$.
I want to compute $\mathbb{P}\{\tau_{uc} = T \mid A_t\}$ when $A_s > A_{uc}$ for all $s\leq t$. I think the probability gotta be equal to zero, but do not know how to derive such a conclusion. Is there anyone who kindly answer this question?