Let $(B_t)$ denote the standard Brownian motion on the interval $[0,1]$. For a given confidence level $\alpha \in (0,1)$ a confidence band on $[0,1]$ is any function $u$ with the property that $$ P(\omega; |B_t(\omega)| < u(t), \quad \forall t\in [0,1])=\alpha. $$ In other words, the probability that a path of the Brownian motion stays within a confidence band is $\alpha$. Additionally the boundary hitting position for those paths leaving the band must be uniformly distributed on $[0,1]$. This condition can be stated using the stopping time $$\tau(\omega) = \inf [ t \in [0,1], |B_t(\omega)|=u(t) ]. $$ Then $\tau $ is the time of the first hitting, and one asks that $\tau$ is uniformly distributed on $[0,1]$ conditionally on the event that $\tau$ is finite.
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The context of the problem is a rather boring one, so will not state it here. The problem itself seem to be non-trivial and interesting.