Classical Wiener space - Revision history

195.37.61.178: /* Definition */

2014-03-06T09:35:11Z

Definition

← Older revision		Revision as of 11:35, 6 March 2014
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	~~{{unreferenced\|date=August 2009}}~~		Hi there, I am Felicidad Oquendo. Interviewing is what I do in my day occupation. Delaware is our birth place. The favorite pastime for him and his children is to drive and now he is trying to earn cash with it.<br><br>Feel free to surf to my web site ... [http://www.Sceltic.at/index.php?mod=users&action=view&id=51301 extended car warranty]
	~~{{Expert-subject\|Mathematics\|date=February 2009}}~~

	~~In [[mathematics]]~~, ~~the '''Malliavin derivative'''~~ is ~~a notion of [[derivative]]~~ in ~~the [[Malliavin calculus]]~~. ~~Intuitively, it~~ is ~~the notion of derivative appropriate to paths in [[classical Wiener space]], which are "usually" not differentiable in the usual sense~~. ~~{{Citation Needed\|date=August 2011}}~~

	~~==Definition==~~
	~~Let <math>H</math> be the [[Cameron-Martin space]],~~ and ~~<math>C_{0}</math> denote [[classical Wiener space]]:~~

	~~:<math>H := \{ f \in W^{1,2} ([0, T]; \mathbb{R}^{n}) \;\|\; f(0) = 0 \} := \{ \text{paths starting at 0 with first derivative in } L^{2} \}</math>;~~

	~~:<math>C_{0} := C_{0} ([0, T]; \mathbb{R}^{n}) := \{ \text{continuous paths starting at 0} \}</math>;~~

	~~By the [[Sobolev_inequality#Sobolev_embedding_theorem\|Sobolev embedding theorem]], <math>H \subset C_0</math>. Let~~
	~~:<math>i : H \to C_{0}</math>~~
	~~denote the [[inclusion map]].~~

	~~Suppose that <math>F : C_{0} \to \mathbb{R}</math> is [[Fréchet derivative\|Fréchet differentiable]]. Then the [[Fréchet derivative]]~~ is ~~a map~~

	~~:<math>\mathrm{D} F : C_{0} \~~to ~~\mathrm{Lin} (C_{0}; \mathbb{R})</math>;~~

	~~i.e., for paths <math>\sigma \in C_{0}</math>, <math>\mathrm{D} F (\sigma)\;</math>~~ is ~~an element of <math>C_{0}^{*}</math>, the [[dual space]]~~ to ~~<math>C_{0}\;</math>~~. ~~Denote by <math>\mathrm{D}_{H} F(\sigma)\;~~<~~/math~~> ~~the [[continuous function\|continuous]] [[linear map]]~~ <~~math~~>~~H \~~to ~~\mathbb{R}</math> defined by~~

	~~:<math>\mathrm{D}_{H} F (\sigma) := \mathrm{D} F (\sigma) \circ i : H \~~to ~~\mathbb{R}, </math>~~

	~~sometimes known as the [[H-derivative\|''H''-derivative]]~~. ~~Now define <math>\nabla_{H} F : C_{0} \to H</math> to be the~~ [~~[adjoint]]{{dn\|date=December 2013}} of <math>\mathrm{D}_{H} F\;</math> in the sense that~~

	~~:<math>\int_0^T \left(\partial_t \nabla_H F(\sigma)\right) \cdot \partial_t h~~ :~~= \langle \nabla_{H} F (\sigma), h \rangle_{H} = \left( \mathrm{D}_{H} F \right) (\sigma) (h) = \lim_{t \to 0} \frac{F (\sigma + t i(h)) - F(\sigma)}{t}<~~/~~math>.~~

	~~Then the '''Malliavin derivative''' <math>\mathrm{D}_{t}<~~/~~math> is defined by~~

	~~:<math>\left( \mathrm{D}_{t} F \right) (\sigma) := \frac{\partial}{\partial t} \left( \left( \nabla_{H} F \right) (\sigma) \right)~~.~~</math>~~

	The [[domain (mathematics)\|domain]] of <math>\mathrm{D}_{t}</math> is the set <math>\mathbf{F}</math> of all Fréchet differentiable real-valued functions on <math>C_{0}\;</math>; the [[codomain]] is <math>L^{2} ([0, T]; \mathbb{R}^{n})</math>.

	~~The '''Skorokhod integral''' <math>\delta\;<~~/~~math> is defined to be the [[adjoint]]{{dn\|date=December 2013}} of the Malliavin derivative:~~

	:<math>\delta := \left( \mathrm{D}_{t} \right)^{} : \operatorname{image} \left( \mathrm{D}_{t} \right) \subseteq L^{2} ([0, T]; \mathbb{R}^{n}) \to \mathbf{F}^{} = \mathrm{Lin} (\mathbf{F}; \mathbb{R}).~~</math>~~

	==~~See also~~==
	*[[H-derivative]]

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Generalizations of the derivative]]~~
	~~[[Category:Stochastic calculus]~~]

128.12.187.216: /* Properties of classical Wiener space */

2013-06-27T03:17:05Z

Properties of classical Wiener space

← Older revision		Revision as of 05:17, 27 June 2013
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	~~Wilber Berryhill~~ is ~~what his wife loves~~ to ~~call him~~ and ~~he completely enjoys this name~~. ~~Office supervising is where her primary income comes from but she's currently applied for another 1~~. ~~Alaska~~ is ~~exactly where he's usually been living~~. ~~Playing badminton~~ is a ~~factor that he~~ is ~~totally addicted~~ to.<br><br>~~My website~~ :: [~~http~~://~~www~~.~~publicpledge~~.~~com~~/~~blogs~~/~~post~~/~~7034 tarot card readings~~]		{{unreferenced\|date=August 2009}}
			{{Expert-subject\|Mathematics\|date=February 2009}}

			In [[mathematics]], the '''Malliavin derivative''' is a notion of [[derivative]] in the [[Malliavin calculus]]. Intuitively, it is the notion of derivative appropriate to paths in [[classical Wiener space]], which are "usually" not differentiable in the usual sense. {{Citation Needed\|date=August 2011}}

			==Definition==
			Let <math>H</math> be the [[Cameron-Martin space]], and <math>C_{0}</math> denote [[classical Wiener space]]:

			:<math>H := \{ f \in W^{1,2} ([0, T]; \mathbb{R}^{n}) \;\|\; f(0) = 0 \} := \{ \text{paths starting at 0 with first derivative in } L^{2} \}</math>;

			:<math>C_{0} := C_{0} ([0, T]; \mathbb{R}^{n}) := \{ \text{continuous paths starting at 0} \}</math>;

			By the [[Sobolev_inequality#Sobolev_embedding_theorem\|Sobolev embedding theorem]], <math>H \subset C_0</math>. Let
			:<math>i : H \to C_{0}</math>
			denote the [[inclusion map]].

			Suppose that <math>F : C_{0} \to \mathbb{R}</math> is [[Fréchet derivative\|Fréchet differentiable]]. Then the [[Fréchet derivative]] is a map

			:<math>\mathrm{D} F : C_{0} \to \mathrm{Lin} (C_{0}; \mathbb{R})</math>;

			i.e., for paths <math>\sigma \in C_{0}</math>, <math>\mathrm{D} F (\sigma)\;</math> is an element of <math>C_{0}^{*}</math>, the [[dual space]] to <math>C_{0}\;</math>. Denote by <math>\mathrm{D}_{H} F(\sigma)\;</math> the [[continuous function\|continuous]] [[linear map]] <math>H \to \mathbb{R}</math> defined by

			:<math>\mathrm{D}_{H} F (\sigma) := \mathrm{D} F (\sigma) \circ i : H \to \mathbb{R}, </math>

			sometimes known as the [[H-derivative\|''H''-derivative]]. Now define <math>\nabla_{H} F : C_{0} \to H</math> to be the [[adjoint]]{{dn\|date=December 2013}} of <math>\mathrm{D}_{H} F\;</math> in the sense that

			:<math>\int_0^T \left(\partial_t \nabla_H F(\sigma)\right) \cdot \partial_t h := \langle \nabla_{H} F (\sigma), h \rangle_{H} = \left( \mathrm{D}_{H} F \right) (\sigma) (h) = \lim_{t \to 0} \frac{F (\sigma + t i(h)) - F(\sigma)}{t}</math>.

			Then the '''Malliavin derivative''' <math>\mathrm{D}_{t}</math> is defined by

			:<math>\left( \mathrm{D}_{t} F \right) (\sigma) := \frac{\partial}{\partial t} \left( \left( \nabla_{H} F \right) (\sigma) \right).</math>

			The [[domain (mathematics)\|domain]] of <math>\mathrm{D}_{t}</math> is the set <math>\mathbf{F}</math> of all Fréchet differentiable real-valued functions on <math>C_{0}\;</math>; the [[codomain]] is <math>L^{2} ([0, T]; \mathbb{R}^{n})</math>.

			The '''Skorokhod integral''' <math>\delta\;</math> is defined to be the [[adjoint]]{{dn\|date=December 2013}} of the Malliavin derivative:

			:<math>\delta := \left( \mathrm{D}_{t} \right)^{} : \operatorname{image} \left( \mathrm{D}_{t} \right) \subseteq L^{2} ([0, T]; \mathbb{R}^{n}) \to \mathbf{F}^{} = \mathrm{Lin} (\mathbf{F}; \mathbb{R}).</math>

			==See also==
			*[[H-derivative]]

			==References==
			{{reflist}}

			[[Category:Generalizations of the derivative]]
			[[Category:Stochastic calculus]]

en>DannyAsher: /* Tightness in classical Wiener space */

2012-08-12T15:26:43Z

Tightness in classical Wiener space

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Wilber Berryhill is what his wife loves to call him and he completely enjoys this name. Office supervising is where her primary income comes from but she's currently applied for another 1. Alaska is exactly where he's usually been living. Playing badminton is a factor that he is totally addicted to.<br><br>My website :: [http://www.publicpledge.com/blogs/post/7034 tarot card readings]