stopping time
Let (ℱt)t∈𝕋 be a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on a set Ω.
A random variable
τ taking values in 𝕋∪{∞} is a stopping time for the filtration (ℱt) if the event {τ≤t}∈ℱt for every t∈𝕋.
Remarks
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The set 𝕋 is the index set
for the time variable t, and the σ-algebra ℱt is the collection
of all events which are observable up to and including time t. Then, the condition that τ is a stopping time means that the outcome of the event {τ≤t} is known at time t.
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In discrete time situations, where 𝕋={0,1,2,…}, the condition that {τ≤t}∈ℱt is equivalent
to requiring that {τ=t}∈ℱt. This is not true for continuous
time cases where 𝕋 is an interval of the real numbers and hence uncountable, due to the fact that σ-algebras are not in general closed under taking uncountable unions of events.
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A random time τ is a stopping time for a stochastic process
(Xt) if it is a stopping time for the natural filtration of X. That is, {τ≤t}∈σ(Xs:s≤t).
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The first time that an adapted process Xt hits a given value or set of values is a stopping time. The inclusion of ∞ into the range of τ is to cover the case where Xt 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.
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The gambler stops playing when his capital is depleted. The number τ=n1 of plays that it takes the gambler to stop is a stopping time.
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The gambler stops playing when his capital reaches $2,000. The number τ=n2 of plays that it takes the gambler to stop is a stopping time.
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The gambler stops playing when his capital either reaches $2,000, or is depleted, which ever comes first. The number τ=min(n1,n2) of plays that it takes the gambler to stop is a stopping time.
Title | stopping time |
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Canonical name | StoppingTime |
Date of creation | 2013-03-22 14:41:13 |
Last modified on | 2013-03-22 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 |