predictable stopping time

A predictable, or previsible stopping time is a random time which is possible to predict just before the event. Letting (t)t+ be a filtrationPlanetmathPlanetmath ( on a measurable spaceMathworldPlanetmathPlanetmath (Ω,), then, a stopping time τ is predictable if there exists an increasing sequence of stopping times τn satisfying the following.

  • τn<τ whenever τ>0.

  • τnτ as n.

The sequence τn is said to announce or foretell τ.

For example, if X is a continuousMathworldPlanetmathPlanetmath adapted process with X0=0, such as Brownian motionMathworldPlanetmath, then the first time τ at which it hits a given level K0 is a predictable stopping time. In this case, if τn is the first time at which X hits the level K(1-1/n), then the sequence τn announces τ.

On the other hand, if X is a Poisson process then the first time τ at which it is nonzero is not predictable. To show this, suppose that τn<τ are stopping times. The fact that Xt-λt is a martingaleMathworldPlanetmath means that Doob’s optional sampling theoremMathworldPlanetmath can be applied, giving 𝔼[Xτn-λτn]=0. Then, Xt=0 for t<τ gives 𝔼[τn]=0. So, τn=0 with probability one, and the sequence τn cannot announce τ.

In discrete time, where the filtration (t) has time t running over the index setMathworldPlanetmathPlanetmath +, then a stopping time is said to be predictable if {τt} is t-1-measurable for every time t=1,2,.

This can be generalized to an arbitrary index set 𝕋, where a stopping time τ:Ω𝕋{} is predictable if there exists an increasing sequence of stopping times τnτ such that τn<τ whenever τ is not equal to a minimal element of 𝕋, and n(τn,τ) contains no elements of 𝕋.

Title predictable stopping time
Canonical name PredictableStoppingTime
Date of creation 2013-03-22 18:37:19
Last modified on 2013-03-22 18:37:19
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Definition
Classification msc 60G40
Classification msc 60G05
Synonym predictable time
Synonym previsible time
Synonym previsible stopping time
Related topic StoppingTime
Related topic PredictableProcess