Base flow (random dynamical systems) - Revision history

en>HitroMilanese: Reverted 1 edit by Aman Janu (talk) to last revision by ClueBot NG. (TW)

2014-02-06T18:34:56Z

Reverted 1 edit by Aman Janu (talk) to last revision by ClueBot NG. (TW)

← Older revision		Revision as of 20:34, 6 February 2014
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	~~In [[mathematics]],~~ the '~~''Itō isometry''', named after [[Kiyoshi Itō]],~~ is ~~a crucial fact about [[Itō calculus\|Itō stochastic integrals]]~~. ~~One of its main applications~~ is ~~to enable the computation of [[variance]]s for [[stochastic processes]]~~.		Alyson Meagher is the name her parents gave her but she doesn't like when individuals use her full name. Her family members lives in Ohio. Credit authorising is where my main earnings comes from. To climb is some thing she would never give up.<br><br>my webpage ... free psychic readings - [http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 just click the up coming article],

	~~Let~~ <~~math~~>~~W : [0, T] \times \Omega \to \mathbb{R}~~<~~/math~~> ~~denote the canonical real~~-~~valued~~ [~~[Wiener process]] defined up to time <math>T > 0</math>, and let <math>X~~ : ~~[0, T] \times \Omega \to \mathbb{R}<~~/~~math> be a stochastic process that is [[adapted process\|adapted]] to the [[filtration (abstract algebra)\|natural filtration]] <math>\mathcal{F}_{*}^{W}<~~/~~math> of the Wiener process~~. ~~Then~~

	~~:<math>\mathbb{E} \left[ \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} \right] = \mathbb{E} \left[ \int_{0}^{T} X_{t}^{2} \, \mathrm{d} t \right],</math>~~

	~~where <math>\mathbb{E}</math> denotes [[expected value\|expectation]] with respect to [[classical Wiener measure]] <math>\gamma</math>~~. ~~In other words, the Itō stochastic integral, as a function, is an [[isometry]] of [[normed vector space]]s with respect to the norms induced by the [[inner product]]s~~

	:<math>( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) = \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \, \mathrm{d} \gamma (\omega)</~~math>~~

	~~and~~

	~~:<math>( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega)~~.~~</math>~~

	== ~~References ==~~

	* {{cite book \| author=Øksendal, Bernt K. \| authorlink=Bernt Øksendal \| title=Stochastic Differential Equations: An Introduction with Applications \| publisher=Springer, ~~Berlin \| year=2003 \| isbn=3-540-04758-1}}~~

	~~{{DEFAULTSORT:Ito Isometry}}~~
	~~[[Category:Stochastic calculus]]~~

en>ClueBot NG: Reverting possible vandalism by Iman Shirwa to version by Widr. False positive? Report it. Thanks, ClueBot NG. (1626527) (Bot)

2013-12-18T23:16:52Z

Reverting possible vandalism by Iman Shirwa to version by Widr. False positive? Report it. Thanks, ClueBot NG. (1626527) (Bot)

← Older revision		Revision as of 01:16, 19 December 2013
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	~~Nestor is~~ the ~~name my mothers and fathers gave me but I don~~'~~t like when people use my complete name~~. One of the ~~things~~ [~~http~~:/~~/product~~-~~dev.younetco.com~~/~~xuannth~~/~~fox_ios1~~/~~index~~.~~php?do~~=/~~profile-700~~/~~info/ extended car warranty~~] ~~car~~ [~~http:~~/~~/www~~.~~angieslist.com/articles/what-does-and-doesnt-your-auto-warranty-cover.htm warranty I] adore most is car warranty climbing and now I have time to take on new things. She works~~ as a ~~monetary officer and she will not~~ [~~http://www.carsdirect.com/car-buying/the-best-car-warranty-coverage-plans-top-used-or-new-auto-service-plans extended auto~~] ~~warranty change it anytime quickly. Her spouse and her chose~~ to [~~http~~:~~//www.sharkz-fifa.de/index.php?mod~~=~~users&action~~=~~view&id=10274 extended auto warranty] car warranty reside in Alabama.~~<br><br>~~Feel free to visit my web blog .~~.. [~~http~~:~~//www.Planetplayschool.com/ActivityFeed/MyProfile/tabid/61/UserId/415/Default.aspx extended auto warranty~~]		In [[mathematics]], the '''Itō isometry''', named after [[Kiyoshi Itō]], is a crucial fact about [[Itō calculus\|Itō stochastic integrals]]. One of its main applications is to enable the computation of [[variance]]s for [[stochastic processes]].

			Let <math>W : [0, T] \times \Omega \to \mathbb{R}</math> denote the canonical real-valued [[Wiener process]] defined up to time <math>T > 0</math>, and let <math>X : [0, T] \times \Omega \to \mathbb{R}</math> be a stochastic process that is [[adapted process\|adapted]] to the [[filtration (abstract algebra)\|natural filtration]] <math>\mathcal{F}_{*}^{W}</math> of the Wiener process. Then

			:<math>\mathbb{E} \left[ \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} \right] = \mathbb{E} \left[ \int_{0}^{T} X_{t}^{2} \, \mathrm{d} t \right],</math>

			where <math>\mathbb{E}</math> denotes [[expected value\|expectation]] with respect to [[classical Wiener measure]] <math>\gamma</math>. In other words, the Itō stochastic integral, as a function, is an [[isometry]] of [[normed vector space]]s with respect to the norms induced by the [[inner product]]s

			:<math>( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) = \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \, \mathrm{d} \gamma (\omega)</math>

			and

			:<math>( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega).</math>

			== References ==

			* {{cite book \| author=Øksendal, Bernt K. \| authorlink=Bernt Øksendal \| title=Stochastic Differential Equations: An Introduction with Applications \| publisher=Springer, Berlin \| year=2003 \| isbn=3-540-04758-1}}

			{{DEFAULTSORT:Ito Isometry}}
			[[Category:Stochastic calculus]]

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes

2009-12-16T11:38:24Z

remove Erik9bot category,outdated, tag and general fixes

New page

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