Martingale representation theorem - Revision history

en>Mark viking: /* Application in finance */ Added wl

2014-02-19T23:28:30Z

Application in finance: Added wl

← Older revision		Revision as of 01:28, 20 February 2014
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	~~{{Calculus}}~~		Myrtle Benny is how I'm called and I really feel comfy when people use the complete name. To do aerobics is a factor that I'm totally addicted to. My working day job is a librarian. Years in the past we moved to North Dakota.<br><br>My blog; at home std test - [http://musical.sehan.ac.kr/?document_srl=2110508 Related Web Page] -
	~~In [[mathematics]], a~~ '~~''time dependent vector field''' is a construction in [[vector calculus]] which generalizes~~ the ~~concept of [[vector field]]s~~. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a [[vector (geometric)\|vector]] to every point in a [[Euclidean space]] or in a [[manifold]].

	~~==Definition==~~
	~~A '''time dependent vector field''' on a manifold ''M''~~ is a ~~map from an open subset <math>\Omega \subset \Bbb{R} \times M</math> on <math>TM</math>~~

	~~:<math>X: \Omega \subset \Bbb{R} \times M \longrightarrow TM</math>~~

	~~::::<math>(t,x) \longmapsto X(t,x)=X_t(x) \in T_xM</math>~~

	~~such~~ that ~~for every <math>(t,x) \in \Omega</math>, <math>X_t(x)</math> is an element of <math>T_xM</math>.~~

	~~For every <math>t \in \Bbb{R}</math> such that the set~~

	~~:<math>\Omega_t=\{x \in M \| (t,x) \in \Omega \} \subset M</math>~~

	~~is [[nonempty]], <math>X_t</math> is a vector field in the usual sense defined on the open set <math>\Omega_t \subset M</math>.~~

	~~==Associated differential equation==~~
	~~Given a time dependent vector field ''X~~'~~' on a manifold ''M'', we can associate~~ to ~~it the following [[differential equation]]:~~

	~~:<math>\frac{dx}{dt}=X(t,x)</math>~~

	~~which is called [[Autonomous system (mathematics)\|nonautonomous]] by definition~~.

	~~==Integral curve==~~

	~~An [[integral curve]] of the equation above (also called an integral curve of ''X'')~~ is a ~~map~~

	~~:<math>\alpha : I \subset \Bbb{R} \longrightarrow M</math>~~

	~~such that <math>\forall t_0 \in I</math>, <math>(t_0,\alpha (t_0))</math> is an element of the [[domain of definition]] of ''X'' and~~

	~~:<math>\frac{d \alpha}{dt} \left~~.~~{\!\!\frac{}{}}\right\|_{t=t_0} =X(t_0,\alpha (t_0))</math>.~~

	~~==Relationship with vector fields in the usual sense==~~
	~~A vector field~~ in the usual sense can be thought of as a time dependent vector field defined on <math>\Bbb{R} \times M</math> even though its value on a point <math>(t,x)</math> does not depend on the component <math>t \in \Bbb{R}</math>.

	~~Conversely, given a time dependent vector field ''X'' defined on <math>\Omega \subset \Bbb{R} \times M</math>,~~ we ~~can associate~~ to it a vector field in the usual sense <math>\tilde{X}</math> on <math>\Omega</math> such that the autonomous differential equation associated to <math>\tilde{X}</math> is essentially equivalent to the nonautonomous differential equation associated to ''X''. ~~It suffices to impose:~~

	~~:<math>\tilde{X}(t,x)=(1,X(t,x))~~<~~/math~~>

	~~for each~~ <~~math~~>~~(t,x) \in \Omega</math>, where we identify <math>T_{(t,x)}(\Bbb{R}\times M)</math> with <math>\Bbb{R}\times T_x M</math>. We can also write it as:~~

	~~:<math> \tilde{X}=\frac{\partial{}}{\partial{t}}+X</math>.~~

	~~To each integral curve of ''X'', we can associate one integral curve of <math>\tilde{X}</math>, and vice versa.~~

	~~==Flow==~~
	~~The [~~[~~Flow (mathematics)\|flow]] of a time dependent vector field ''X'', is the unique differentiable map~~

	:~~<math>F:D(X) \subset \Bbb{R} \times \Omega \longrightarrow M</math>~~

	~~such that for every <math>(t_0,x) \in \Omega</math>,~~

	~~:<math>t \longrightarrow F(t,t_0,x)</math>~~

	~~is the integral curve of ''X'' <math>\alpha</math> that verifies <math>\alpha (t_0) = x</math>.~~

	~~===Properties===~~
	~~We define <math>F_{t,s}</math> as <math>F_{t,s}(p)=F(t,s,p)<~~/~~math>~~

	~~#If <math>(t_1,t_0,p) \in D(X)</math> and <math>(t_2,t_1,F_{t_1,t_0}(p)) \in D(X)<~~/~~math> then <math>F_{t_2,t_1} \circ F_{t_1,t_0}(p)=F_{t_2,t_0}(p)</math>~~
	~~#<math>\forall t,s</math>, <math>F_{t,s}</math> is a [[diffeomorphism]] with [[Inverse function\|inverse]] <math>F_{s,t}</math>~~.

	~~==Applications==~~
	~~Let ''X'' and ''Y'' be smooth time dependent vector fields and <math>F</math> the flow of ''X''~~. ~~The following identity can be proved:~~

	~~:<math>\frac{d}{dt} \left~~ .~~{\!\!\frac{}{}}\right\|_{t=t_1} (F^_{t,t_0} Y_t)_p = \left( F^_{t_1,t_0} \left( [X_{t_1},Y_{t_1}] + \frac{d}{dt} \left .{\!\!\frac{}{}}\right\|_{t=t_1} Y_t \right) \right)_p</math>~~

	~~Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that <math>\eta<~~/~~math> is a smooth time dependent tensor field:~~

	~~:<math>\frac{d}{dt} \left .{\!\!\frac{}{}}\right\|_{t=t_1} (F^_{t,t_0} \eta_t)_p = \left( F^_{t_1,t_0} \left( \mathcal{L}_{X_{t_1}}\eta_{t_1} + \frac{d}{dt} \left .{\!\!\frac{}{}}\right\|_{t~~=~~t_1} \eta_t \right) \right)_p</math>~~

	~~This last identity is useful to prove the [[Darboux theorem]~~].

	~~==References==~~
	* Lee, John M., ''Introduction to Smooth Manifolds'', Springer-~~Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.~~

	~~[[Category:Differential geometry]]~~
	~~[[Category:Vector calculus]]~~

134.60.117.253: /* Application in finance */

2013-11-07T08:01:20Z

Application in finance

← Older revision		Revision as of 10:01, 7 November 2013
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	~~Hi there~~, ~~I am Sophia~~. ~~Her family life~~ in ~~Alaska but her spouse wants them to transfer~~. ~~Distributing production~~ is ~~where my main earnings comes~~ from ~~and~~ it'~~s some thing~~ I ~~truly appreciate~~. ~~The preferred pastime for him and his kids~~ is to ~~play lacross~~ and ~~he would by no means give it up~~.<br><br>~~Stop by my homepage - real psychic~~ (~~[http~~://~~www~~.~~indosfriends~~.~~com~~/~~profile-253~~/~~info~~/ ~~indosfriends~~.~~com~~])		{{Calculus}}
			In [[mathematics]], a '''time dependent vector field''' is a construction in [[vector calculus]] which generalizes the concept of [[vector field]]s. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a [[vector (geometric)\|vector]] to every point in a [[Euclidean space]] or in a [[manifold]].

			==Definition==
			A '''time dependent vector field''' on a manifold ''M'' is a map from an open subset <math>\Omega \subset \Bbb{R} \times M</math> on <math>TM</math>

			:<math>X: \Omega \subset \Bbb{R} \times M \longrightarrow TM</math>

			::::<math>(t,x) \longmapsto X(t,x)=X_t(x) \in T_xM</math>

			such that for every <math>(t,x) \in \Omega</math>, <math>X_t(x)</math> is an element of <math>T_xM</math>.

			For every <math>t \in \Bbb{R}</math> such that the set

			:<math>\Omega_t=\{x \in M \| (t,x) \in \Omega \} \subset M</math>

			is [[nonempty]], <math>X_t</math> is a vector field in the usual sense defined on the open set <math>\Omega_t \subset M</math>.

			==Associated differential equation==
			Given a time dependent vector field ''X'' on a manifold ''M'', we can associate to it the following [[differential equation]]:

			:<math>\frac{dx}{dt}=X(t,x)</math>

			which is called [[Autonomous system (mathematics)\|nonautonomous]] by definition.

			==Integral curve==

			An [[integral curve]] of the equation above (also called an integral curve of ''X'') is a map

			:<math>\alpha : I \subset \Bbb{R} \longrightarrow M</math>

			such that <math>\forall t_0 \in I</math>, <math>(t_0,\alpha (t_0))</math> is an element of the [[domain of definition]] of ''X'' and

			:<math>\frac{d \alpha}{dt} \left.{\!\!\frac{}{}}\right\|_{t=t_0} =X(t_0,\alpha (t_0))</math>.

			==Relationship with vector fields in the usual sense==
			A vector field in the usual sense can be thought of as a time dependent vector field defined on <math>\Bbb{R} \times M</math> even though its value on a point <math>(t,x)</math> does not depend on the component <math>t \in \Bbb{R}</math>.

			Conversely, given a time dependent vector field ''X'' defined on <math>\Omega \subset \Bbb{R} \times M</math>, we can associate to it a vector field in the usual sense <math>\tilde{X}</math> on <math>\Omega</math> such that the autonomous differential equation associated to <math>\tilde{X}</math> is essentially equivalent to the nonautonomous differential equation associated to ''X''. It suffices to impose:

			:<math>\tilde{X}(t,x)=(1,X(t,x))</math>

			for each <math>(t,x) \in \Omega</math>, where we identify <math>T_{(t,x)}(\Bbb{R}\times M)</math> with <math>\Bbb{R}\times T_x M</math>. We can also write it as:

			:<math> \tilde{X}=\frac{\partial{}}{\partial{t}}+X</math>.

			To each integral curve of ''X'', we can associate one integral curve of <math>\tilde{X}</math>, and vice versa.

			==Flow==
			The [[Flow (mathematics)\|flow]] of a time dependent vector field ''X'', is the unique differentiable map

			:<math>F:D(X) \subset \Bbb{R} \times \Omega \longrightarrow M</math>

			such that for every <math>(t_0,x) \in \Omega</math>,

			:<math>t \longrightarrow F(t,t_0,x)</math>

			is the integral curve of ''X'' <math>\alpha</math> that verifies <math>\alpha (t_0) = x</math>.

			===Properties===
			We define <math>F_{t,s}</math> as <math>F_{t,s}(p)=F(t,s,p)</math>

			#If <math>(t_1,t_0,p) \in D(X)</math> and <math>(t_2,t_1,F_{t_1,t_0}(p)) \in D(X)</math> then <math>F_{t_2,t_1} \circ F_{t_1,t_0}(p)=F_{t_2,t_0}(p)</math>
			#<math>\forall t,s</math>, <math>F_{t,s}</math> is a [[diffeomorphism]] with [[Inverse function\|inverse]] <math>F_{s,t}</math>.

			==Applications==
			Let ''X'' and ''Y'' be smooth time dependent vector fields and <math>F</math> the flow of ''X''. The following identity can be proved:

			:<math>\frac{d}{dt} \left .{\!\!\frac{}{}}\right\|_{t=t_1} (F^_{t,t_0} Y_t)_p = \left( F^_{t_1,t_0} \left( [X_{t_1},Y_{t_1}] + \frac{d}{dt} \left .{\!\!\frac{}{}}\right\|_{t=t_1} Y_t \right) \right)_p</math>

			Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that <math>\eta</math> is a smooth time dependent tensor field:

			:<math>\frac{d}{dt} \left .{\!\!\frac{}{}}\right\|_{t=t_1} (F^_{t,t_0} \eta_t)_p = \left( F^_{t_1,t_0} \left( \mathcal{L}_{X_{t_1}}\eta_{t_1} + \frac{d}{dt} \left .{\!\!\frac{}{}}\right\|_{t=t_1} \eta_t \right) \right)_p</math>

			This last identity is useful to prove the [[Darboux theorem]].

			==References==
			* Lee, John M., ''Introduction to Smooth Manifolds'', Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.

			[[Category:Differential geometry]]
			[[Category:Vector calculus]]

en>Zfeinst: /* Statement of the theorem */ wikilink for 'predictable process'

2011-10-15T01:50:54Z

Statement of the theorem: wikilink for 'predictable process'

New page

Hi there, I am Sophia. Her family life in Alaska but her spouse wants them to transfer. Distributing production is where my main earnings comes from and it's some thing I truly appreciate. The preferred pastime for him and his kids is to play lacross and he would by no means give it up.<br><br>Stop by my homepage - real psychic ([http://www.indosfriends.com/profile-253/info/ indosfriends.com])