Stopping Time in layman's words

  • #1
The Tortoise-Man
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5
I have a question about intuitive meaning of stopping time in stochastics. A random variable ##\tau: \Omega \to \mathbb {N} \cup \{ \infty \}## is called a stopping time with resp to a discrete filtration ##(\mathcal F_n)_{n \in \mathbb {N}_0}##of ##\Omega ## , if for any ##n \in \mathbb{N}## holds## \{ \omega \in \Omega : T(\omega)=n\} \in \mathcal F_n ##

What does it mean on an intuitive level? Eg in wikipedia it is said that ntuitively, this condition means that the "decision" of whether to stop at time ##t=n##must be based only on the information present at time ##t=n##, not on any future information.

In other words, this means to me that at ## t=n ## we "know" (ie can exactly decide) whether ##T(\omega)## has reached ## n ## or not for every ##\omega \in \Omega##.

I don't understand the idea behind it. Why does it read as an intuitive interpretation of the formal condition ##\{ \omega \in \Omega : T(\omega)=n\} \in \mathcal F_n##? Cannot we simply following this intuitive interpretation in order to check for arbitrary ##\omega \in \Omega## holds ##T(\omega)=n## just pick it, evaluate in ##T## and check "by hand" without imposing this technical condition## \{ \omega \in \Omega : T(\omega)=n\} \in \mathcal F_n ##
? Why is it crucial?

In other words, why and in which sense the technical condition ## \{ \omega \in \Omega : T(\omega)=n\} \in \mathcal F_n ## "translates" into this intuitive picture?
Indeed let's recall, according to the intuitive interpretation of a filtration on a stochastic space ##\Omega##, one can interpret for every ##n## the ##\mathcal F_n \subset \mathcal F_{\infty}:= \mathcal F## as a collection of events that are "measurable already at time ##t=n##", i.e. at time ##t=n ## the stochastic evaluation ##P(A)## only makes “sense” if ##A \in \mathcal F_n ##, otherwise we cannot evaluate yet other events ##A## at time ##n## stochastically as long as ##A \not \in \mathcal F_n ##.

In other words, at time ##t=n## we can assign a certain probability of occurrence to an event ##A \in \mathcal F_n ##, but of course we cannot judge exactly (i.e. as ##P(A) = 0## or ##1##, otherwise nothing) whether ##A## occurred or not, right?
Therefore I not understand now the technical condition ## \{ \omega \in \Omega : T(\omega)=n\} \in \mathcal F_n ## translates in layman's statement that "at time ##t=n## we "know" if ##T## has passed ##n##. What we can calculate at this point is a probability due to above, but we cannot decide it exactly in deterministic sense, right?
 
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  • #2
This seems like such an abstract definition for which you are looking for an intuitive interpretation. I certainly am having trouble with it. Is there a reason that you want to start with such an abstract presentation?
 
  • #3
FactChecker said:
This seems like such an abstract definition for which you are looking for an intuitive interpretation. I certainly am having trouble with it. Is there a reason that you want to start with such an abstract presentation?

So if your question is why I started with "such abstract definition" the answer is simply because that this is exactly the standard definition of stopping time and my motivation is to understand it on intuitive level, what seems apparently to be possible as I explained above (... cp with wikipedia or nearly every site you can find googleing "stopping time intuition"). But I still not understood it, that's my motivation :)
 
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  • #4
Consider the following direction:

"Turn left at the second junction after the gas station."

You pass a gas station.
You continue on and reach a junction. Is this the correct turn? No.
Continuing on, you reach another junction. Is this the correct turn? Yes.

Now instead consider this direction:

"Turn left at the second-to-last junction before the gas station."

You reach a junction with a left turn. You have not yet reached a gas station. Is this the correct turn? You can't know.
Continuing on, you reach a junction with a left turn, You have not yet reached a gas station. Is this the correct turn? You can't know.
Continuing on, you pass a gas station. You realize you should have turned at the first junction you passed.

The first direction defines the correct turn by means of a stopping time. The second does not.
 
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  • #5
The Tortoise-Man said:
So if your question is why I started with "such abstract definition" the answer is simply because that this is exactly the standard definition of stopping time and my motivation is to understand it on intuitive level, what seems apparently to be possible as I explained above (... cp with wikipedia or nearly every site you can find googleing "stopping time intuition"). But I still not understood it, that's my motivation :)
Sorry. I didn't realize that it was a standard definition for stopping time. I have only seen it in concrete examples and never discussed in such an abstract way. I will leave this to others.
 
  • #6
pasmith said:
Consider the following direction:

"Turn left at the second junction after the gas station."

You pass a gas station.
You continue on and reach a junction. Is this the correct turn? No.
...

Thanks, that's verry illuminative example! Since my primary concern was to understand the slogan why the technical condition ## \{ \omega \in \Omega : T(\omega)=n\} \in \mathcal F_n ## translates on intuitive level into slogan that at time ##t=n## we know if ##T## has happened at ##n##, lets fill this example with math.

Assume we work with discrete time steps, ie ##t=0,1,2,...##. Let the filtration ##(F_n)_n## of ##\Omega## be induced by iid random variables ##X_m: \Omega \to \{0,1\}## defined by if at ##m##-th time unit ##\omega## sees a station, then ##1##, and ##0## else.
Is this model reasonable?

If ok far, let me try to implement your examples: first example was directed by random variable ##T: \Omega \to \mathbb{N}## with ##T(\omega)=## "minimal ##m## with ##X_m(\omega)=1## (which is stopping time as you said).On the other hand the second (non example) ##S: \Omega \to \mathbb{N}## with ##S(\omega)=## "minimal ##m## with ##X_{m+1}(\omega)=1## (ie about a future event; so not a ST).

Is this attept to model your examples correct so far? If yes, then I have still a question about this condition ## \{ \omega \in \Omega : T(\omega)=n\} \in \mathcal F_n##:Why imposing this condition can be phrased informally exactly as that "we know" at time ##n## if ##T## happend there or not?
 
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  • #7
Let me add this explanation which provides an excellent transition from intuitive (see pasmith's very enlightening examples above) to formal interpretation of the essence of stopping times!
 

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