optional process

Suppose we are given a filtrationPlanetmathPlanetmath (http://planetmath.org/FiltrationOfSigmaAlgebras) (β„±)tβˆˆπ•‹ on a measurable spaceMathworldPlanetmathPlanetmath (Ξ©,β„±). A stochastic processMathworldPlanetmath is said to be adapted if Xt is β„±t-measurable for every time t in the index setMathworldPlanetmathPlanetmath 𝕋. For an arbitrary, uncountable, index set π•‹βŠ†β„, this property is too restrictive to be useful. Instead, we can impose measurability conditions on X considered as a map from 𝕋×Ω to ℝ. 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 𝕋 is assumed to be a closed subset of ℝ 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 Οƒ-algebra, π’ͺ, on 𝕋×Ω generated by the right-continuous and adapted processes is called the optional Οƒ-algebra. Then, a process is optional if and only if it is π’ͺ-measurable.

Alternatively, the optional Οƒ-algebra may be defined as

π’ͺ=σ⁒({[T,∞):T⁒ is a stopping time}).

Here, [T,∞) is a stochastic interval, consisting of the pairs (t,Ο‰)βˆˆπ•‹Γ—Ξ© such that T⁒(Ο‰)≀t. In continuous-time, the equivalence of these two definitions for π’ͺ does require mild conditions on the filtration β€” it is enough for β„±t to be universally complete.

In the discrete-time case where the index set 𝕋 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