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en>Bgwhite: Do general fixes and cleanup. - using AWB (10480)

2014-10-11T06:44:26Z

Do general fixes and cleanup. - using AWB (10480)

← Older revision		Revision as of 08:44, 11 October 2014
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71.102.130.240 at 18:09, 4 March 2014

2014-03-04T18:09:33Z

← Older revision		Revision as of 20:09, 4 March 2014
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	~~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~~.		The writer is known as Irwin Wunder but it's not the most masucline title out there. His family members life in South Dakota but his spouse wants them to move. To gather coins is a factor that I'm completely addicted to. Managing individuals has been his day job for a while.<br><br>Also visit my blog post ... [http://www.smylestream.org/groups/solid-advice-in-relation-to-candida/ std home test]

	~~==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]]~~

	~~{{probability-stub}}~~
	~~{{mathanalysis-stub}}~~

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

2012-12-18T09:35:22Z

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

← Older revision		Revision as of 11:35, 18 December 2012
Line 1:		Line 1:
	~~Signwriter Russ Meece from Osoyoos~~, ~~has several interests including scrapbooking~~, ~~como~~ [~~http~~://~~ganhedinheiro~~.~~comoganhardinheiro101~~.~~com~~/ ~~ganhar dinheiro~~] ~~na internet~~ and ~~maintain~~ a ~~journal~~. ~~Gets motivation through travel~~ and ~~just spent 8 months at Ningaloo Coast~~.		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]]

			{{probability-stub}}
			{{mathanalysis-stub}}

en>Crisco 1492: Cleaned up using AutoEd

2011-06-03T23:46:17Z

Cleaned up using AutoEd

New page

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