What is meant by a stopping time?

Solution 1:

You have some event, which you typically don't know when occurs, but that can/will occur some time in the future. The time that this event occurs is random, and it is a stopping time if, at any point in time, you know whether the event has occurred or not.

A few quick examples.

1) Your own (a stopping time): Let $\tau$ denote the time that I'm ruined (i.e. when I have no money left). At any time, I know whether I am ruined or not. For instance, I am not ruined right now. I don't know when ruin occurs, or if it will occur at all, but if it does, I will know.

2) Parking (not a stopping time): Suppose I am driving along a very long road, and that I'm looking for the parking spot which is furthest towards the other end of the road (call this "the last parking spot"). I pass by available spots along the way, but at any time, I never know if I have passed the last free parking spot. Why? I could just have passed some empty spot, but I cannot see if there are more empty spots later on, and I wouldn't know if the spot that I just passed was the last one or not.

3) My birthday this year (a stopping time): This is a deterministic stopping time. At any time, I know whether or not my birthday has occurred this year. In fact, I know exactly when my birthday occurs, which makes this a non-typical stopping time in the sense that it is deterministic.

Solution 2:

$τ = \min\{n : X_n = 0\}$ is the first $n$ such that $X_n =0.$ i.e. the first time that the process hits zero, as you said.

A non-stopping time would the first time $n$ such that $X_n = \max X_j$. The reason being that at a given time you don't know where the process will go next.

Solution 3:

You seem to understand the concept pretty well. Just like you, I would have said that the time of ruin is $τ = \min\{n : X_n \leq 0\}$ instead of $τ = \min\{n : X_n = 0\}$. But in this precise example, the time of ruin is the first time that you have exactly 0.

The concept of stopping time is closely related to that of filtration of a stochastic process. In other words, $\tau $ is a stopping time if the event $\lbrace\tau \leq n\rbrace$ is measurable, with respect to the filtration you're using, which is usually $\mathcal{F}_n=\sigma(X_0,\dots,X_n)$.