What is meant by a stopping time?

Question

TL;DR:

• is a stopping time some sort of event, or is it a point in discrete time, or something else entirely

• what is an example of something which is not a stopping time?

• is my understanding of the concepts and definitions below correct?

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I am having difficulty understanding what a stopping time is.

The definition I am provided with is as follows: A random time $τ$ is called a stopping time if for any $n$, one can decide whether the event $\{τ ≤ n\}$ (and hence the complementary event $\{τ > n\}$) has occurred by observing the first n variables $X_1, X_2, . . . , X_n$.

We are then given an example: Time of ruin is a stopping time.

$τ = \min\{n : X_n = 0\}$. $\{τ > n\} = \{X_1 ≥ 0, X_2 ≥ 0, . . . , X_n > 0\}$.

I don't quite understand what this is supposed to tell us.

random time $τ$ is called a stopping time if for any $n$, one can decide whether the event $\{τ ≤ n\}$ has occurred by observing the first n variables $X_1, X_2, . . . , X_n$

When they say the event $\{ τ \leq n\}$ they are referring to some specific time, are they not? e.g $τ = 1$ or maybe $τ = 4$ as long as $τ \leq n$

Is this correct?

So then if we know the event $ \{τ \leq n\}$ has or has not occurred, we can conclude whether the complementary even $τ > n$ has occurred.

If this is fine so far, then I have issues with the example.

Time of ruin is stopping time, $τ = \min\{n : X_n = 0\}$

Firstly, time of ruin to me means at a point where you have $0$ or a negative balance of some sort of asset (For a gambler, no more money to gamble with, for a business owner, no more cash to pay expenses or obligations) - is this correct?

In that case Time of ruin should occur when $X_n \leq 0$, correct?

IF that is fine, then continuing, what does

$τ = \min\{n : X_n = 0\}$ mean? This is not the same as $τ = \min\{n,X_n\}$, is it? What is it trying to say? I read it as, the minimum of $n$, such that $X_n = 0$

So it's saying $τ$ is the first point at which we are ruined?

Is my understanding all correct? Can someone provide me with an example of what is NOT a stopping time? Does a "stopping time" refer to a type of event?

Answer 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.

Answer 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.

Answer 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 $\mathcal{F}_n=\sigma(X_0,\dots,X_n)$.

Answer 4

Stopping time is sth we know whether it happens or not at each step (based on all information till now). e.g. in the tossing coin gambling, each time we toss a coin, we know whether the gambler loses (lose all bet), wins (win all bet) or neither.

The example of not stopping time, may include max/best/furthest, which usually needs information of whole process to get.