stopping time


Let (t)t𝕋 be a filtrationPlanetmathPlanetmath (http://planetmath.org/FiltrationOfSigmaAlgebras) on a set Ω. A random variableMathworldPlanetmath τ taking values in 𝕋{} is a stopping time for the filtration (t) if the event {τt}t for every t𝕋.

Remarks

  • The set 𝕋 is the index setMathworldPlanetmathPlanetmath for the time variable t, and the σ-algebra t is the collectionMathworldPlanetmath 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 {τt} is known at time t.

  • In discrete time situations, where 𝕋={0,1,2,}, the condition that {τt}t is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to requiring that {τ=t}t. This is not true for continuousPlanetmathPlanetmath 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.

  • A random time τ is a stopping time for a stochastic processMathworldPlanetmath (Xt) if it is a stopping time for the natural filtration of X. That is, {τt}σ(Xs:st).

  • 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.

  • 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. 1.

    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.

  2. 2.

    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.

  3. 3.

    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
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