Part of Doobs Optional Stopping theorem is that for an almost surely finite stopping time $T$ and a Martingale $(X_n)_{n \in \mathbb{N_0}}$ it is enough to assume that T is bounded to get:
$E[X_T] = E[X_0]$
We defined bounded as: There exists $N \in \mathbb{N}_0$ such that $P(T \leq N) = 1$
I encountered many examples of a.s. finite stopping times that are unbounded like e.g.:
$T = min\{n \in \mathbb{N}|S_n = A\}$ for the simple random walk $S_n$ and some $A \in \mathbb{Z}$.
Please give an example of a bounded stopping time and explain what the difference is w.r.t. structure. (with structure I mean e.g.: first hitting times of random walks are usually unbounded)
As John Dawkins commented: If $T$ is any stopping time, then for $n \in \mathbb{N}$ fixed, the $\min(n, T)$ is a bounded stopping time. An example for that would be to gamble until you won a certain amount or until the casino closes.
A constant stopping time would of course also be bounded. I'm still hoping for a less trivial example though, if there is one.
Thank you!