Bounded and finite stopping times

Question

Can someone tell me the difference between bounded stopping times and finite stopping times? They seem to be the same thing.

Answer 1

A stopping time is just a random variable. For a random variable $X$, we say that $X$ is bounded almost surely if $\mathrm{P}[X < M] = 1$ for some $M \in \mathbb{R}$, while $X$ is finite almost surely if $\mathrm{P}[X < \infty] = 1$.

Clearly bounded a.s. implies finite a.s., but not the converse. One good example is $X \sim N(0,1)$, where $\mathrm{P}[X < M] < 1$ for all $M \in \mathbb{R}$, but $\mathrm{P}[X = \infty] = 0$.