optional process


Suppose we are given a filtrationPlanetmathPlanetmath (http://planetmath.org/FiltrationOfSigmaAlgebras) (ā„±)tāˆˆš•‹ on a measurable spaceMathworldPlanetmathPlanetmath (Ī©,ā„±). A stochastic processMathworldPlanetmath is said to be adapted if Xt is ā„±t-measurable for every time t in the index setMathworldPlanetmathPlanetmath š•‹. For an arbitrary, uncountable, index set š•‹āŠ†ā„, this property is too restrictive to be useful. Instead, we can impose measurability conditions on X considered as a map from š•‹Ć—Ī© to ā„. For instance, we could require X to be progressively measurable, but that is still too weak a condition for many purposes. A stronger condition is for X to be optional. The index set š•‹ is assumed to be a closed subset of ā„ in the following definition.

The class of optional processes forms the smallest set containing all adapted and right-continuous processes, and which is closed under taking limits of sequences of processes.

The Ļƒ-algebra, š’Ŗ, on š•‹Ć—Ī© generated by the right-continuous and adapted processes is called the optional Ļƒ-algebra. Then, a process is optional if and only if it is š’Ŗ-measurable.

Alternatively, the optional Ļƒ-algebra may be defined as

š’Ŗ=Ļƒā¢({[T,āˆž):Tā¢Ā is a stopping time}).

Here, [T,āˆž) is a stochastic interval, consisting of the pairs (t,Ļ‰)āˆˆš•‹Ć—Ī© such that Tā¢(Ļ‰)ā‰¤t. In continuous-time, the equivalence of these two definitions for š’Ŗ does require mild conditions on the filtration ā€” it is enough for ā„±t to be universally complete.

In the discrete-time case where the index set š•‹ countableMathworldPlanetmath, then the definitions above imply that a process Xt is optional if and only if it is adapted.

Title optional process
Canonical name OptionalProcess
Date of creation 2013-03-22 18:37:34
Last modified on 2013-03-22 18:37:34
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Definition
Classification msc 60G07
Related topic ProgressivelyMeasurableProcess
Related topic PredictableProcess
Defines optional