Stopping rule

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Example of a stopping time: a hitting time of Brownian motion
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Example of a stopping time: a hitting time of Brownian motion

In probability theory, in particular in the study of stochastic processes, a stopping time is a specific type of "random time".

The theory of stopping rules and stopping times can be analysed in probability and statistics, notably in the optional stopping theorem. This says that under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.

[edit] Definition

A stopping time with respect to a sequence of random variables X1, X2, ... is a random variable τ with the property that for each t, the occurrence or non-occurrence of the event τ = t depends only on the values of X1, X2, ..., Xt, and furthermore Pr(τ < ∞) = 1. Stopping times occur in decision theory, in which a stopping rule is characterized as a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some time.

Another, more general, definition may be given in terms of a filtered sigma algebra: let (I, \leq) be an ordered index set (often [0,T] or [0, + \infty)), let (\Omega, \mathcal{F}, \mathbb{P}) be a probability space, and let \mathcal{F}_{*} := \{ F_{i} | i \in I \} be a filtration of \mathcal{F}. Then a random variable \tau : \Omega \to I is called a stopping time for the filtration \mathcal{F}_{*} if the event \tau \leq i is \mathcal{F}_{i}-measurable for each i \in I:

\forall i \in I, \{ \omega \in \Omega | \tau (\omega) \leq i \} \in \mathcal{F}_{i}.

As noted above, it is frequently required that τ be almost surely finite, although some authors omit this requirement.

[edit] Example

As an example, consider a gambler playing roulette, starting with $100:

  • Playing until she either runs out of money or has played 500 games is a stopping rule.
  • Playing until she doubles her money (borrowing if necessary if she goes into debt) is not a stopping rule, as there is a positive probability that she will never double her money.
  • Playing until she either doubles her money or runs out of money is a stopping rule, even though there is potentially no limit to the number of games she plays, since the probability that she stops in a finite time is 1.
  • Playing until she is the maximum amount ahead she will ever be is not a stopping rule and does not provide a stopping time, as it requires information about the future as well as the present and past.

[edit] See also

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