A common Brownian Motion quiz ([[Zhou Xinfeng]]): Given a simple BM, what's the probability that it hits 3 before it hits -5?
This is actually identical to the BM with upper and lower boundaries. The BM walker stops when it hits either boundary. We know it eventually stops. At that stopping time, the walker is either at 3 or -5 but which is more likely?
Ultimately, we rely on the optional stopping theorem – At the stopping time, the martingale's value is a random variable and its expectation is equal to the initial value.