stopping time
Let ${({\mathcal{F}}_{t})}_{t\in \mathbb{T}}$ be a filtration^{} (http://planetmath.org/FiltrationOfSigmaAlgebras) on a set $\mathrm{\Omega}$.
A random variable^{} $\tau $ taking values in $\mathbb{T}\cup \{\mathrm{\infty}\}$ is a stopping time for the filtration $({\mathcal{F}}_{t})$ if the event $\{\tau \le t\}\in {\mathcal{F}}_{t}$ for every $t\in \mathbb{T}$.
Remarks

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The set $\mathbb{T}$ is the index set^{} for the time variable $t$, and the $\sigma $algebra ${\mathcal{F}}_{t}$ is the collection^{} of all events which are observable up to and including time $t$. Then, the condition that $\tau $ is a stopping time means that the outcome of the event $\{\tau \le t\}$ is known at time $t$.

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In discrete time situations, where $\mathbb{T}=\{0,1,2,\mathrm{\dots}\}$, the condition that $\{\tau \le t\}\in {\mathcal{F}}_{t}$ is equivalent^{} to requiring that $\{\tau =t\}\in {\mathcal{F}}_{t}$. This is not true for continuous^{} time cases where $\mathbb{T}$ is an interval of the real numbers and hence uncountable, due to the fact that $\sigma $algebras are not in general closed under taking uncountable unions of events.

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A random time $\tau $ is a stopping time for a stochastic process^{} $({X}_{t})$ if it is a stopping time for the natural filtration of $X$. That is, $\{\tau \le t\}\in \sigma ({X}_{s}:s\le t)$.

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The first time that an adapted process ${X}_{t}$ hits a given value or set of values is a stopping time. The inclusion of $\mathrm{\infty}$ into the range of $\tau $ is to cover the case where ${X}_{t}$ never hits the given values.

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Stopping time is often used in gambling, when a gambler stops the betting process when he reaches a certain goal. The time it takes to reach this goal is generally not a deterministic one. Rather, it is a random variable depending on the current result of the bet, as well as the combined information from all previous bets.
Examples. A gambler has $1,000 and plays the slot machine at $1 per play.

1.
The gambler stops playing when his capital is depleted. The number $\tau ={n}_{1}$ of plays that it takes the gambler to stop is a stopping time.

2.
The gambler stops playing when his capital reaches $2,000. The number $\tau ={n}_{2}$ of plays that it takes the gambler to stop is a stopping time.

3.
The gambler stops playing when his capital either reaches $2,000, or is depleted, which ever comes first. The number $\tau =\mathrm{min}({n}_{1},{n}_{2})$ of plays that it takes the gambler to stop is a stopping time.
Title  stopping time 

Canonical name  StoppingTime 
Date of creation  20130322 14:41:13 
Last modified on  20130322 14:41:13 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  11 
Author  gel (22282) 
Entry type  Definition 
Classification  msc 60K05 
Classification  msc 60G40 
Related topic  DoobsOptionalSamplingTheorem 
Related topic  PredictableStoppingTime 