stopped process

A stochastic processMathworldPlanetmath (Xt)t∈𝕋 defined on a measurable spaceMathworldPlanetmathPlanetmath (Ω,ℱ) can be stopped at a random time τ:Ω→𝕋∪{∞}. The resulting stopped process is denoted by Xτ,


The random time τ used is typically a stopping time.

If the process Xt has left limits ( for every t∈𝕋, then it can alternatively be stopped just before the time τ, resulting in the pre-stopped process

Xτ-≡{Xt,if ⁢t<τ,Xτ-,if ⁢t≥τ.

Stopping is often used to enforce boundedness or integrability constraints on a process. For example, if B is a Brownian motionMathworldPlanetmath and τ is the first time at which |Bτ| hits some given positive value, then the stopped process Bτ will be a continuousPlanetmathPlanetmath and boundedPlanetmathPlanetmathPlanetmath martingaleMathworldPlanetmath. It can be shown that many properties of stochastic processes, such as the martingale property, are stable under stopping at any stopping time τ. On the other hand, a pre-stopped martingale need not be a martingale.

For continuous processes, stopping and pre-stopping are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath procedures. If τ is the first time at which |Xτ|≥K, for any given real number K, then the pre-stopped process Xτ- will be uniformly bounded. However, for some noncontinuous processes it is not possible to find a stopping time τ>0 making Xτ into a uniformly bounded process. For example, this is the case for any Levy process ( with unboundedPlanetmathPlanetmath jump distributionPlanetmathPlanetmath.

Stopping is used to generalize properties of stochastic processes to obtain the related localized property. See, for example, local martingalesPlanetmathPlanetmath.

Title stopped process
Canonical name StoppedProcess
Date of creation 2013-03-22 18:37:38
Last modified on 2013-03-22 18:37:38
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Definition
Classification msc 60G40
Classification msc 60G05
Synonym optional stopping
Defines pre-stopped process
Defines prestopped process