71.90.103.119: /* Sample calculations */

2014-02-02T00:38:32Z

Sample calculations

← Older revision		Revision as of 02:38, 2 February 2014
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	~~Wilber Berryhill~~ is the ~~title his mothers~~ and ~~fathers gave him~~ and ~~he totally digs~~ that ~~name~~. ~~Distributing manufacturing~~ is ~~how he tends~~ to ~~make~~ a ~~living~~. ~~To perform domino~~ is ~~something I truly enjoy performing~~. ~~Her family lives~~ in ~~Ohio~~.<br><br>~~my web page~~; ~~spirit messages ([http~~://~~www~~.~~seekavideo~~.~~com~~/~~playlist~~/~~2199~~/~~video/ relevant web site~~])		In [[mathematics]], an '''integration by parts operator''' is a [[linear operator]] used to formulate [[integration by parts]] formulae; the most interesting examples of integration by parts operators occur in infinite-dimensional settings and find uses in [[stochastic analysis]] and its applications.

			==Definition==

			Let ''E'' be a [[Banach space]] such that both ''E'' and its [[continuous dual space]] ''E''<sup>∗</sup> are [[separable space]]s; let ''μ'' be a [[Borel measure]] on ''E''. Let ''S'' be any (fixed) [[subset]] of the class of functions defined on ''E''. A linear operator ''A'' : ''S'' → ''L''<sup>2</sup>(''E'', ''μ''; '''R''') is said to be an '''integration by parts operator''' for ''μ'' if

			:<math>\int_{E} \mathrm{D} \varphi(x) h(x) \, \mathrm{d} \mu(x) = \int_{E} \varphi(x) (A h)(x) \, \mathrm{d} \mu(x)</math>

			for every [[smooth function\|''C''<sup>1</sup> function]] ''φ'' : ''E'' → '''R''' and all ''h'' ∈ ''S'' for which either side of the above equality makes sense. In the above, D''φ''(''x'') denotes the [[Fréchet derivative]] of ''φ'' at ''x''.

			==Examples==

			* Consider an [[abstract Wiener space]] ''i'' : ''H'' → ''E'' with abstract Wiener measure ''γ''. Take ''S'' to be the set of all ''C''<sup>1</sup> functions from ''E'' into ''E''<sup>∗</sup>; ''E''<sup>∗</sup> can be thought of as a subspace of ''E'' in view of the inclusions

			::<math>E^{} \xrightarrow{i^{}} H^{*} \cong H \xrightarrow{i} E.</math>

			:For ''h'' ∈ ''S'', define ''Ah'' by

			::<math>(A h)(x) = h(x) x - \mathrm{trace}_{H} \mathrm{D} h(x).</math>

			:This operator ''A'' is an integration by parts operator, also known as the [[divergence]] operator; a proof can be found in Elworthy (1974).

			* The [[classical Wiener space]] ''C''<sub>0</sub> of [[continuous function\|continuous paths]] in '''R'''<sup>''n''</sup> starting at zero and defined on the [[interval (mathematics)\|unit interval]] [0, 1] has another integration by parts operator. Let ''S'' be the collection

			::<math>S = \left\{ \left. h \colon C_{0} \to L_{0}^{2, 1} \right\| h \mbox{ is bounded and non-anticipating} \right\},</math>

			:i.e., all [[bounded function\|bounded]], [[adapted process\|adapted]] processes with [[absolutely continuous]] sample paths. Let ''φ'' : ''C''<sub>0</sub> → '''R''' be any ''C''<sup>1</sup> function such that both ''φ'' and D''φ'' are bounded. For ''h'' ∈ ''S'' and ''λ'' ∈ '''R''', the [[Girsanov theorem]] implies that

			::<math>\int_{C_{0}} \varphi (x + \lambda h(x)) \, \mathrm{d} \gamma(x) = \int_{C_{0}} \varphi(x) \exp \left( \lambda \int_{0}^{1} \dot{h}_{s} \cdot \mathrm{d} x_{s} - \frac{\lambda^{2}}{2} \int_{0}^{1} \| \dot{h}_{s} \|^{2} \, \mathrm{d} s \right) \, \mathrm{d} \gamma(x).</math>

			:Differentiating with respect to ''λ'' and setting ''λ'' = 0 gives

			::<math>\int_{C_{0}} \mathrm{D} \varphi(x) h(x) \, \mathrm{d} \gamma(x) = \int_{C_{0}} \varphi(x) (A h) (x) \, \mathrm{d} \gamma(x),</math>

			:where (''Ah'')(''x'') is the [[Itō integral]]

			::<math>\int_{0}^{1} \dot{h}_{s} \cdot \mathrm{d} x_{s}.</math>

			:The same relation holds for more general ''φ'' by an approximation argument; thus, the Itō integral is an integration by parts operator and can be seen as an infinite-dimensional divergence operator. This is the same result as the [[Clark-Ocone theorem#Integration by parts on Wiener space\|integration by parts formula derived from the Clark-Ocone theorem]].

			==References==

			* {{cite book
			\| last = Bell
			\| first = Denis R.
			\| title = The Malliavin calculus
			\| publisher = Dover Publications Inc.
			\| location = Mineola, NY
			\| year = 2006
			\| pages = x+113
			\| isbn = 0-486-44994-7
			}} {{MathSciNet\|id=2250060}} (See section 5.3)
			* {{cite book
			\| last = Elworthy
			\| first = K. David
			\| chapter = Gaussian measures on Banach spaces and manifolds
			\| title = Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol. II
			\| pages = 151–166
			\| publisher = Internat. Atomic Energy Agency
			\| address = Vienna
			\| year = 1974
			}} {{MathSciNet\|id=0464297}}

			[[Category:Integral calculus]]
			[[Category:Measure theory]]
			[[Category:Operator theory]]
			[[Category:Stochastic processes]]

en>Life of Riley: Removed "it must be noted that" in accordance with WP:NOTED.

2012-08-04T21:39:32Z

Removed "it must be noted that" in accordance with WP:NOTED.

New page

Wilber Berryhill is the title his mothers and fathers gave him and he totally digs that name. Distributing manufacturing is how he tends to make a living. To perform domino is something I truly enjoy performing. Her family lives in Ohio.<br><br>my web page; spirit messages ([http://www.seekavideo.com/playlist/2199/video/ relevant web site])

Direct integration of a beam - Revision history

71.90.103.119: /* Sample calculations */

en>Life of Riley: Removed "it must be noted that" in accordance with WP:NOTED.