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The Tortoise-Man

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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?

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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