Level set (data structures)

2012-12-18T09:35:22Z

95.166.111.77: Two author publication was credited to only the first author. Thus, the second author was added.

In [[mathematics]], '''progressive measurability''' is a property in the theory of [[stochastic processes]]. A progressively measurable process, while defined quite technically, is important because it implies the [[stopped process]] is [[measurable]]. Being progressively measurable is a strictly stronger property than the notion of being an [[adapted process]].<ref name="Karatzas">{{cite book|last=Karatzas|first=Ioannis|last2=Shreve|first2=Steven|year=1991|title=Brownian Motion and Stochastic Calculus|publisher=Springer|edition=2nd|isbn=0-387-97655-8|pages=4–5}}</ref> Progressively measurable processes are important in the theory of [[Itō integral]]s.

==Definition==
Let
* <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [[probability space]];
* <math>(\mathbb{X}, \mathcal{A})</math> be a [[measurable space]], the ''state space'';
* <math>\{ \mathcal{F}_{t} | t \geq 0 \}</math> be a [[Filtration_(abstract_algebra)|filtration]] of the [[sigma algebra]] <math>\mathcal{F}</math>;
* <math>X : [0, \infty) \times \Omega \to \mathbb{X}</math> be a [[stochastic process]] (the index set could be <math>[0, T]</math> or <math>\mathbb{N}_{0}</math> instead of <math>[0, \infty)</math>).

The process <math>X</math> is said to be '''progressively measurable'''<ref name=Pasc>Pascucci, Andrea (2011) ''PDE and Martingale Methods in Option Pricing''. Berlin: Springer {{Page needed|date=August 2011}}</ref> (or simply '''progressive''') if, for every time <math>t</math>, the map <math>[0, t] \times \Omega \to \mathbb{X}</math> defined by <math>(s, \omega) \mapsto X_{s} (\omega)</math> is <math>\mathrm{Borel}([0, t]) \otimes \mathcal{F}_{t}</math>-[[Measurable_function|measurable]]. This implies that <math>X</math> is <math> \mathcal{F}_{t} </math>-adapted.<ref name="Karatzas" />

A subset <math>P \subseteq [0, \infty) \times \Omega</math> is said to be '''progressively measurable''' if the process <math>X_{s} (\omega) := \chi_{P} (s, \omega)</math> is progressively measurable in the sense defined above, where <math>\chi_{P}</math> is the [[indicator function]] of <math>P</math>. The set of all such subsets <math>P</math> form a sigma algebra on <math>[0, \infty) \times \Omega</math>, denoted by <math>\mathrm{Prog}</math>, and a process <math>X</math> is progressively measurable in the sense of the previous paragraph if, and only if, it is <math>\mathrm{Prog}</math>-measurable.

==Properties==
* It can be shown<ref name="Karatzas" /> that <math>L^2 (B)</math>, the space of stochastic processes <math>X : [0, T] \times \Omega \to \mathbb{R}^n</math> for which the [[Ito integral]]
:: <math>\int_0^T X_t \, \mathrm{d} B_t </math>
: with respect to [[Brownian motion]] <math>B</math> is defined, is the set of [[equivalence class]]es of <math>\mathrm{Prog}</math>-measurable processes in <math>L^2 ([0, T] \times \Omega; \mathbb{R}^n)\,</math>.
* Every adapted process with left- or [[Continuous function#Directional continuity|right-continuous]] paths is progressively measurable. Consequently, every adapted process with [[càdlàg]] paths is progressively measurable.<ref name="Karatzas" />
* Every measurable and adapted process has a progressively measurable modification.<ref name="Karatzas" />

==References==
{{reflist}}

[[Category:Stochastic processes]]
[[Category:Measure theory]]

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{{mathanalysis-stub}}

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Level set (data structures)