SABR volatility model - Revision history

208.252.128.20: Fix original paper link

2014-08-19T19:07:38Z

Fix original paper link

← Older revision		Revision as of 21:07, 19 August 2014
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	In the mathematical area of [[order theory]], a '''completely distributive lattice''' is a [[complete lattice]] in which arbitrary [[join (lattice theory)\|join]]s [[distributivity (order theory)\|distribute]] over arbitrary [[meet (lattice theory)\|meet~~]]s.~~		Nice to meet you, my name is Refugia. I am a meter reader. What I adore performing is performing ceramics but I haven't made a dime with it. South Dakota is where me and my husband live and my family members loves it.<br><br>Here is my blog post :: [http://www.animecontent.com/blog/349119 std testing at home]

	~~Formally~~, ~~a complete lattice ''L'' is said to be '''completely distributive''' if, for any doubly indexed family~~
	~~{''x''<sub>''j'',''k''</sub> \| ''j'' in ''J'', ''k'' in ''K''<sub>''j''</sub>} of ''L'', we have~~
	~~: <math>\begin{align}\bigwedge_{j\in J}\bigvee_{k\in K_j} x_{j,k} =~~
	~~\bigvee_{f\in F}\bigwedge_{j\in J} x_{j,f(j)}\end{align}</math>~~
	~~where ''F''~~ is ~~the set of [[choice function]]s ''f'' choosing for each index ''j'' of ''J'' some index ''f''(''j'') in ''K''<sub>''j''</sub>~~.~~<ref name="DaveyPriestley">B. A. Davey and H. A. Priestey, ''Introduction to Lattices and Order'' 2nd Edition, Cambridge University Press, 2002, ISBN 0-521-78451-4</ref>~~

	~~Complete distributivity is~~ a ~~self-dual property, i.e. [[Duality (order theory)\|dualizing]] the above statement yields the same class of complete lattices~~.~~<ref name="DaveyPriestley"/>~~

	~~==Alternative characterizations==~~

	~~Various different characterizations exist. For example, the following~~ is an equivalent law that avoids the use of choice functions{{Citation needed\|date=February 2007}}. For any set ''S'' of sets, we define the set ''S''<sup>#</sup> to be the set of all subsets ''X'' of the complete lattice that have non-empty intersection with all members of ''S''. We then can define complete distributivity via the statement

	~~: <math>\begin{align}\bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\{ \bigwedge Z \mid Z\in S^\# \}\end{align}</math>~~

	~~The operator ( )<sup>#</sup> might be called the '''crosscut operator'''. This version of complete distributivity only implies the original notion when admitting the [[Axiom of Choice]].~~

	~~<!-- This isn~~'t ~~valid. See talk.~~
	~~However, the latter version is always equivalent to the statement:~~

	~~: <math>\begin{align}\bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\bigcap S\end{align}</math>~~

	~~for all sets ''S'' of subsets of~~ a ~~complete lattice~~.
	~~-->~~

	~~==Properties==~~
	~~In addition, it~~ is ~~known that the following statements are equivalent for any complete lattice ''L''{{Citation needed\|date=February 2007}}:~~

	* ''L'' is completely distributive.
	* ''L'' can be embedded into a direct product of chains [0,1] by an [[order embedding]] that preserves arbitrary meets and ~~joins~~.
	* Both ''L'' and its dual order ''L''<~~sup~~>op<~~/sup~~> ~~are [[continuous poset]]s.~~

	~~Direct products of [0,1], i.e. sets of all functions from some set ''X'' to [0,1] ordered [[pointwise order\|pointwise]], are also called ''cubes''.~~

	~~==Free completely distributive lattices==<!-- This section~~ is ~~linked from [[Completely distributive lattice]]. See [[WP~~:~~MOS#Section management]] -->~~
	~~Every [[partially ordered set\|poset]] ''C'' can be [[Complete lattice#Completion\|completed]] in a completely distributive lattice.~~

	A completely distributive lattice ''L'' is called the '''free completely distributive lattice over a poset ''C''''' if and only if there is an [[order embedding]] <math>\phi:C\rightarrow L</math> such that for every completely distributive lattice ''M'' and [[monotonic function]] <math>f:C\rightarrow M</math>, there is a unique [[Complete lattice#Morphisms of complete lattices\|complete homomorphism]] <math>f^_\phi:L\rightarrow M</math> satisfying <math>f=f^_\phi\circ\phi</math>. For every poset ''C'', the free completely distributive lattice over a poset ''C'' exists and is unique up to isomorphism.<ref name="Morris04">Joseph M. Morris, ''[http://www.~~springerlink~~.com/~~content~~/~~nfqh0l29f3unrlwh Augmenting Types with Unbounded Demonic and Angelic Nondeterminacy]'', Mathematics of Program Construction, LNCS 3125, 274-288, 2004</ref>~~

	This is an instance of the concept of [[free object]]. Since a set ''X'' can be considered as a poset with the discrete order, the above result guarantees the existence of the free completely distributive lattice over the set ''X''.

	~~==Examples==~~
	* The [[unit interval]] [0,1], ordered in the natural way, is a completely distributive lattice.<ref name="Raney52">G. N. Raney, ''Completely distributive complete lattices'', Proceedings of the [[American Mathematical Society]], 3: 677 - 680, 1952.</ref>
	**More generally, any [[Total order#Completeness\|complete chain]] is a completely distributive lattice.<ref name="hopenwasser90">Alan Hopenwasser, ''Complete Distributivity'', Proceedings of Symposia in Pure Mathematics, 51(1), 285 - 305, 1990.</ref>
	* The [[power set]] lattice <math>(\mathcal{P}(X),\subseteq)</math> for any set ''X'' is a completely distributive lattice.<ref name="DaveyPriestley"/>
	* For every poset ''C'', there is a ''free completely distributive lattice over C''.<ref name="Morris04"/> See the section on [[Completely distributive lattice#Free completely distributive lattices\|Free completely distributive lattices]] above.

	~~==See also==~~
	* [[Glossary of order theory]]
	* [[Distributive lattice]]

	~~==References==~~

	~~<references/>~~

	~~[[Category:Order theory]~~]

en>Cydebot: Robot - Speedily moving category Options to :Category:Options (finance) per CFDS.

2013-11-09T08:05:35Z

Robot - Speedily moving category Options to Category:Options (finance) per CFDS.

← Older revision		Revision as of 10:05, 9 November 2013
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	The ~~name~~ of the ~~author~~ is ~~Figures but~~ it's ~~not~~ the ~~most masucline title out~~ there. ~~Puerto Rico~~ is ~~exactly where he's usually been residing but she needs~~ to ~~transfer simply because of her family~~. [http://~~blogzaa~~.com/~~blogs~~/~~post~~/~~9199~~ over the ~~counter std test~~] ~~favorite hobby~~ for ~~my kids and me~~ is ~~to perform baseball but I haven't produced~~ a ~~dime with it~~. ~~My day occupation~~ is a ~~meter reader~~.		In the mathematical area of [[order theory]], a '''completely distributive lattice''' is a [[complete lattice]] in which arbitrary [[join (lattice theory)\|join]]s [[distributivity (order theory)\|distribute]] over arbitrary [[meet (lattice theory)\|meet]]s.

			Formally, a complete lattice ''L'' is said to be '''completely distributive''' if, for any doubly indexed family
			{''x''<sub>''j'',''k''</sub> \| ''j'' in ''J'', ''k'' in ''K''<sub>''j''</sub>} of ''L'', we have
			: <math>\begin{align}\bigwedge_{j\in J}\bigvee_{k\in K_j} x_{j,k} =
			\bigvee_{f\in F}\bigwedge_{j\in J} x_{j,f(j)}\end{align}</math>
			where ''F'' is the set of [[choice function]]s ''f'' choosing for each index ''j'' of ''J'' some index ''f''(''j'') in ''K''<sub>''j''</sub>.<ref name="DaveyPriestley">B. A. Davey and H. A. Priestey, ''Introduction to Lattices and Order'' 2nd Edition, Cambridge University Press, 2002, ISBN 0-521-78451-4</ref>

			Complete distributivity is a self-dual property, i.e. [[Duality (order theory)\|dualizing]] the above statement yields the same class of complete lattices.<ref name="DaveyPriestley"/>

			==Alternative characterizations==

			Various different characterizations exist. For example, the following is an equivalent law that avoids the use of choice functions{{Citation needed\|date=February 2007}}. For any set ''S'' of sets, we define the set ''S''<sup>#</sup> to be the set of all subsets ''X'' of the complete lattice that have non-empty intersection with all members of ''S''. We then can define complete distributivity via the statement

			: <math>\begin{align}\bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\{ \bigwedge Z \mid Z\in S^\# \}\end{align}</math>

			The operator ( )<sup>#</sup> might be called the '''crosscut operator'''. This version of complete distributivity only implies the original notion when admitting the [[Axiom of Choice]].

			<!-- This isn't valid. See talk.
			However, the latter version is always equivalent to the statement:

			: <math>\begin{align}\bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\bigcap S\end{align}</math>

			for all sets ''S'' of subsets of a complete lattice.
			-->

			==Properties==
			In addition, it is known that the following statements are equivalent for any complete lattice ''L''{{Citation needed\|date=February 2007}}:

			* ''L'' is completely distributive.
			* ''L'' can be embedded into a direct product of chains [0,1] by an [[order embedding]] that preserves arbitrary meets and joins.
			* Both ''L'' and its dual order ''L''<sup>op</sup> are [[continuous poset]]s.

			Direct products of [0,1], i.e. sets of all functions from some set ''X'' to [0,1] ordered [[pointwise order\|pointwise]], are also called ''cubes''.

			==Free completely distributive lattices==<!-- This section is linked from [[Completely distributive lattice]]. See [[WP:MOS#Section management]] -->
			Every [[partially ordered set\|poset]] ''C'' can be [[Complete lattice#Completion\|completed]] in a completely distributive lattice.

			A completely distributive lattice ''L'' is called the '''free completely distributive lattice over a poset ''C''''' if and only if there is an [[order embedding]] <math>\phi:C\rightarrow L</math> such that for every completely distributive lattice ''M'' and [[monotonic function]] <math>f:C\rightarrow M</math>, there is a unique [[Complete lattice#Morphisms of complete lattices\|complete homomorphism]] <math>f^_\phi:L\rightarrow M</math> satisfying <math>f=f^_\phi\circ\phi</math>. For every poset ''C'', the free completely distributive lattice over a poset ''C'' exists and is unique up to isomorphism.<ref name="Morris04">Joseph M. Morris, ''[http://www.springerlink.com/content/nfqh0l29f3unrlwh Augmenting Types with Unbounded Demonic and Angelic Nondeterminacy]'', Mathematics of Program Construction, LNCS 3125, 274-288, 2004</ref>

			This is an instance of the concept of [[free object]]. Since a set ''X'' can be considered as a poset with the discrete order, the above result guarantees the existence of the free completely distributive lattice over the set ''X''.

			==Examples==
			* The [[unit interval]] [0,1], ordered in the natural way, is a completely distributive lattice.<ref name="Raney52">G. N. Raney, ''Completely distributive complete lattices'', Proceedings of the [[American Mathematical Society]], 3: 677 - 680, 1952.</ref>
			**More generally, any [[Total order#Completeness\|complete chain]] is a completely distributive lattice.<ref name="hopenwasser90">Alan Hopenwasser, ''Complete Distributivity'', Proceedings of Symposia in Pure Mathematics, 51(1), 285 - 305, 1990.</ref>
			* The [[power set]] lattice <math>(\mathcal{P}(X),\subseteq)</math> for any set ''X'' is a completely distributive lattice.<ref name="DaveyPriestley"/>
			* For every poset ''C'', there is a ''free completely distributive lattice over C''.<ref name="Morris04"/> See the section on [[Completely distributive lattice#Free completely distributive lattices\|Free completely distributive lattices]] above.

			==See also==
			* [[Glossary of order theory]]
			* [[Distributive lattice]]

			==References==

			<references/>

			[[Category:Order theory]]

en>Btyner: +cat

2012-07-11T03:00:05Z

+cat

New page

The name of the author is Figures but it's not the most masucline title out there. Puerto Rico is exactly where he's usually been residing but she needs to transfer simply because of her family. [http://blogzaa.com/blogs/post/9199 over the counter std test] favorite hobby for my kids and me is to perform baseball but I haven't produced a dime with it. My day occupation is a meter reader.