Let $B$ be a Brownian motion and define $L=\sup \{ t \leq 1 : B_t = 0 \}$.

My question is:

Why is $L$ an interesting random time?

Durrett's probability book proves something about the distribution of $L$ and also defines an analogous random variable and proves similar results for discrete simple random walks. But why do we care about this? When is $L$ useful? Why do I care about the last time I hit zero before $t=1$?


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