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stopping time (Definition)

Let $ \lbrace X_n \rbrace$ be a stochastic process adopted to the filtration $ \lbrace \mathcal{F}_n \rbrace$ of sub-$ \sigma$-fields $ \mathcal{F}_n$ of $ \mathcal{F}$. A random variable $ \tau$ whose values lie in $ \lbrace 0,1,\ldots \rbrace \cup \lbrace \infty \rbrace$ is a stopping time for the filtration $ \lbrace \mathcal{F}_n \rbrace$ if the event $ \lbrace \tau=n \rbrace \in \mathcal{F}_n$ for every finite $ n\ge0$.

Remarks

  • 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.
  • The $ \sigma$-field at time $ n$, or $ \mathcal{F}_n$, intuitively, is just a collection of events, or, “information” on all possible outcomes that are available up to, and including, the $ n$th bet.
  • The inclusion of $ \infty$ in the range of $ \tau$ is to add a case where there is no stopping time; the process continues into infinity.
  • The stopping time $ \tau$ can be alternatively characterized as a non-negative integer-valued random variable, such that the event $ \lbrace \tau = n \rbrace$ it is independent of $ X_{n+1}, X_{n+2}, \ldots$.

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=\operatorname{min}(n_1,n_2)$ of plays that it takes the gambler to stop is a stopping time.



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See Also: Doob's optional sampling theorem

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Cross-references: machine, independent, infinity, range, inclusion, outcomes, collection, information, current, finite, event, random variable, filtration, stochastic process
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This is version 3 of stopping time, born on 2004-10-04, modified 2005-02-28.
Object id is 6294, canonical name is StoppingTime.
Accessed 6705 times total.

Classification:
AMS MSC60G40 (Probability theory and stochastic processes :: Stochastic processes :: Stopping times; optimal stopping problems; gambling theory)
 60K05 (Probability theory and stochastic processes :: Special processes :: Renewal theory)

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