Let $S_t$ be standard Brownian motion (or a Wiener process) in one dimension. How do I formally derive the expected time that $S_t$ will leave a given interval $[-x, y]$ for some $x, y > 0$, given that $S_0 = 0$?
I suspect that the answer is $xy$ since Brownian motion is the limit of a simple random walk and $xy$ would be the answer for a simple random walk I think, but how do you derive this formally using the definition of a Brownian motion as a limit?