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2014-07-23T22:01:06Z

References: Task 5: Fix CS1 deprecated coauthor parameter errors

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	~~In probability theory~~, ~~a real valued [[stochastic process\|process]] ''X'' is called a '''semimartingale''' if~~ it ~~can be decomposed as the sum of a [[local martingale]] and an adapted finite-variation process~~.		Hi there, I am Alyson Pomerleau and I think it seems fairly good when you say it. Some time in the past she chose to reside in Alaska and her mothers and fathers live close by. Office supervising is what she does for a living. The [http://jplusfn.gaplus.kr/xe/qna/78647 free psychic readings] preferred hobby for him and his children is fashion and he'll be beginning some thing else along with it.<br><br>Have a look at my blog - authentic [http://ltreme.com/index.php?do=/profile-127790/info/ phone psychic] readings - [http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 chorokdeul.co.kr] -
	~~Semimartingales are "good integrators", forming~~ the ~~largest class of processes with respect~~ to ~~which the [[Itō integral]]~~ and ~~the [[Stratonovich integral]] can be defined~~.

	~~The class of semimartingales~~ is ~~quite large (including,~~ for example, all continuously differentiable processes, [[Wiener process\|Brownian motion]] and [[Poisson process]]es). [[Martingale_(probability_theory)#Submartingales_and_supermartingales\|Submartingales]] and [[Martingale_(probability_theory)#Submartingales_and_supermartingales\|supermartingales]] together represent a ~~subset of the semimartingales~~.

	~~==Definition==~~
	~~A real valued process ''X'' defined on the~~ [[filtration (mathematics)#Measure theory\|filtered probability space]] (Ω,''F'',(''F''<sub>''t''</sub>)<sub>''t'' ≥ 0</sub>,P) is called a '''semimartingale''' if it can be decomposed as
	:~~<math>X_t = M_t + A_t</math>~~
	~~where ''M'' is a [[local martingale]] and ''A'' is a [[càdlàg]] [[adapted process]] of locally [[bounded variation]].~~

	~~An '''R'''<sup>''n''<~~/~~sup>-valued process ''X'' = (''X''<sup>1<~~/~~sup>,…,''X''<sup>''n''</sup>) is a semimartingale if each of its components ''X''<sup>''i''</sup> is a semimartingale~~.

	~~==Alternative definition==~~
	First, the simple [[predictable process]]es are defined to be linear combinations of processes of the form ''H''<sub>''t''</sub> = ''A''1<sub>{''t'' > ''T''}</sub> for stopping times ''T'' and ''F''<sub>''T''</sub> -measurable random variables ''A''. ~~The integral ''H'' · ''X'' for any such simple predictable process ''H'' and real valued process ''X'' is~~
	~~:<math>H\cdot X_t\equiv 1_{\{t>T\}}A(X_t-X_T).<~~/~~math>~~
	~~This is extended to all simple predictable processes by the linearity of ''H'' · ''X'' in ''H''.~~

	~~A real valued process ''X'' is a semimartingale if it is càdlàg, adapted, and for every ''t'' ≥ 0,~~

	~~:<math>\left\{H\cdot X_t:H{\rm\ is\ simple\ predictable\ and\ }\|H\|\le 1\right\}<~~/~~math>~~

	~~is bounded in probability. The Bichteler-Dellacherie Theorem states that these two definitions are equivalent {{Harv\|Protter\|2004\|p=144}}.~~

	~~==Examples==~~
	* Adapted and continuously differentiable processes are finite variation processes, and hence are semimartingales.
	* [[Wiener process\|Brownian motion]] is a semimartingale.
	* All càdlàg [[Martingale (probability theory)\|martingales]], submartingales and supermartingales are semimartingales.
	* [[Itō calculus#Itō processes\|Itō processes]], which satisfy a stochastic differential equation of the form ''dX'' = ''σdW'' + ''μdt'' are semimartingales. Here, ''W'' is a Brownian motion and ''σ, μ'' are adapted processes.
	* Every [[Lévy process]] is a semimartingale.

	~~Although most continuous and adapted processes studied in the literature are semimartingales, this is not always the case.~~

	* [[Fractional Brownian motion]] with Hurst parameter ''H'' ≠ 1/~~2 is not a semimartingale.~~

	~~==Properties==~~

	* The semimartingales form the largest class of processes for which the [[Itō calculus\|Itō integral]] can be defined.
	* Linear combinations of semimartingales are semimartingales.
	* Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the [[Stochastic calculus\|Itō integral]].
	* The [[quadratic variation]~~] exists~~ for ~~every semimartingale.~~
	* The class of semimartingales is closed under [[stopped process\|optional stopping]], [[Stopping time#Localization\|localization]], [[change of time]] and ~~absolutely continuous [[Absolutely_continuous#Absolute_continuity_of_measures\|change of measure]].~~
	* If ''X'' is ~~an '''R'''<sup>''m''</sup> valued semimartingale~~ and '~~'f'' is a twice continuously differentiable function from '''R'''<sup>''m''</sup> to '''R'''<sup>''n''</sup>, then ''f''(''X'') is a semimartingale. This is a consequence of [[Itō's lemma]].~~
	* The property of being a semimartingale is preserved under shrinking the filtration. ~~More precisely, if ''X'' is a semimartingale with respect to the filtration ''F''~~<~~sub~~>t<~~/sub~~>~~, and is adapted with respect to the subfiltration ''G''<sub>t</sub>, then ''X'' is~~ a ~~''G''<sub>t</sub>~~-~~semimartingale.~~
	* (Jacod's Countable Expansion) The property of being a semimartingale is preserved under enlarging the filtration by a countable set of disjoint sets. Suppose that ''F''<sub>t</~~sub> is a filtration, and ''G''<sub>t<~~/~~sub> is the filtration generated by ''F''<sub>t</sub> and a countable set of disjoint measurable sets~~. ~~Then, every ''F''<sub>t</sub>-semimartingale is also a ''G''<sub>t<~~/~~sub>-semimartingale. {{Harv\|Protter\|2004\|p=53}}~~

	~~==Semimartingale decompositions==~~
	~~By definition, every semimartingale is a sum of a local martingale and a finite variation process. However, this decomposition is not unique.~~

	~~===Continuous semimartingales===~~
	~~A continuous semimartingale uniquely decomposes as ''X'' = ''M'' + ''A'' where ''M'' is a continuous local martingale and ''A'' is a continuous finite variation process starting at zero~~. ~~{{Harv\|Rogers\|Williams\|1987\|p~~=~~358}}~~

	~~For example, if ''X'' is an Itō process satisfying the stochastic differential equation d''X''<sub>t<~~/~~sub> = σ<sub>t</sub> d''W''<sub>t</sub> + ''b''<sub>t</sub> dt, then~~
	~~:<math>M_t=X_0+\int_0^t\sigma_s\,dW_s,\ A_t=\int_0^t b_s\,ds.</math>~~

	~~===Special semimartingales===~~
	A special semimartingale is a real valued process ''X'' with the decomposition ''X'' = ''M'' + ''A'', where ''M'' is a local martingale and ''A'' is a predictable finite variation process starting at zero. If this decomposition exists, then it is unique up to a P-~~null set.~~

	~~Every special semimartingale is a semimartingale. Conversely, a semimartingale is a special semimartingale if and only if the process ''X''<sub>t<~~/~~sub><sup>*<~~/~~sup> &equiv; sup<sub>''s'' ≤ ''t''</sub> \|X<sub>''s''</sub>\| is [[Stopping time#Localization\|locally integrable~~]~~] {{Harv\|Protter\|2004\|p=130}}.~~

	~~For example, every continuous semimartingale is a special semimartingale, in which case ''M'' and ''A'' are both continuous processes.~~

	~~===Purely discontinuous semimartingales===~~
	~~A semimartingale is called purely discontinuous if its quadratic variation~~ [~~''X''] is a pure jump process,~~
	:~~<math>[X]_t=\sum_{s\le t}\Delta X_s^2<~~/~~math>~~.
	~~Every adapted finite variation process is a purely discontinuous semimartingale~~. ~~A continuous process is a purely discontinuous semimartingale if and only if it is an adapted finite variation process.~~

	Then, every semimartingale has the unique decomposition ''X'' = ''M'' + ''A'' where ''M'' is a continuous local martingale and ''A'' is a purely discontinuous semimartingale starting at zero. The local martingale ''M'' - ''M''<sub>0</~~sub> is called the continuous martingale part of ''X'', and written as ''X''<sup>c</sup> ({{Harvnb\|He\|Wang\|Yan\|1992\|p=209}}; {{Harvnb\|Kallenberg\|2002\|p=527}})~~.

	~~In particular, if ''X'' is continuous, then ''M'' and ''A'' are continuous.~~

	==~~Semimartingales on a manifold==~~

	The concept of semimartingales, and the associated theory of stochastic calculus, extends to processes taking values in a [[differentiable manifold]]. A process ''X'' on the manifold ''M'' is a semimartingale if ''f''(''X'') is a semimartingale for every smooth function ''f'' from ''M'' to '''R'''. ~~{{Harv\|Rogers\|1987\|p=24}} Stochastic calculus for semimartingales on general manifolds requires the use of the [[Stratonovich integral]]~~.

	~~==See also==~~
	*[[Sigma-martingale]]

	~~==References==~~
	*{{Citation\|last=He\|first=Sheng-wu\|last2=Wang\|first2=Jia-gang\|last3=Yan\|first3=Jia-an\|year=1992\|title=Semimartingale Theory and Stochastic Calculus\|publisher=Science Press, CRC Press Inc.\|isbn= 0-8493-7715-3}}
	*{{Citation\|last=Kallenberg\|first=Olav\|year=2002\|title=Foundations of Modern Probability\|edition=2nd\|publisher=Springer\|isbn=0-387-95313-2}}
	*{{Citation\|last=Protter\|first=Philip E.\|year=2004\|title=Stochastic Integration and Differential Equations\|publisher=Springer\|edition=2nd\|isbn=3-~~540-00313-4}}~~
	*{{Citation\|last=Rogers\|first=L.C.G.\|last2=Williams\|first2=David\|year=1987\|volume=2\|title=Diffusions, Markov Processes, and Martingales\|publisher=John Wiley & Sons Ltd\|isbn=0-471-91482-7}}

	~~{{Stochastic processes}}~~

	~~[[Category:Stochastic processes]]~~
	~~[[Category:Martingale theory]]~~

en>Suslindisambiguator at 18:00, 14 September 2013

2013-09-14T18:00:31Z

← Older revision		Revision as of 20:00, 14 September 2013
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	~~Claude~~ is ~~her title~~ and ~~she totally digs~~ that ~~title~~. ~~The factor I adore~~ most ~~bottle tops gathering~~ and ~~now I have~~ time to ~~take on new issues~~. ~~For years she~~'s ~~been living in Kansas~~. ~~I am~~ a ~~manufacturing~~ and ~~distribution officer~~.<br><br>~~my page~~; ~~[http~~:/~~/pchelpnow~~.~~biz~~/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/60/~~userId~~/~~48445~~/~~Default~~.~~aspx pchelpnow~~.~~biz~~]		In probability theory, a real valued [[stochastic process\|process]] ''X'' is called a '''semimartingale''' if it can be decomposed as the sum of a [[local martingale]] and an adapted finite-variation process.
			Semimartingales are "good integrators", forming the largest class of processes with respect to which the [[Itō integral]] and the [[Stratonovich integral]] can be defined.

			The class of semimartingales is quite large (including, for example, all continuously differentiable processes, [[Wiener process\|Brownian motion]] and [[Poisson process]]es). [[Martingale_(probability_theory)#Submartingales_and_supermartingales\|Submartingales]] and [[Martingale_(probability_theory)#Submartingales_and_supermartingales\|supermartingales]] together represent a subset of the semimartingales.

			==Definition==
			A real valued process ''X'' defined on the [[filtration (mathematics)#Measure theory\|filtered probability space]] (Ω,''F'',(''F''<sub>''t''</sub>)<sub>''t'' ≥ 0</sub>,P) is called a '''semimartingale''' if it can be decomposed as
			:<math>X_t = M_t + A_t</math>
			where ''M'' is a [[local martingale]] and ''A'' is a [[càdlàg]] [[adapted process]] of locally [[bounded variation]].

			An '''R'''<sup>''n''</sup>-valued process ''X'' = (''X''<sup>1</sup>,…,''X''<sup>''n''</sup>) is a semimartingale if each of its components ''X''<sup>''i''</sup> is a semimartingale.

			==Alternative definition==
			First, the simple [[predictable process]]es are defined to be linear combinations of processes of the form ''H''<sub>''t''</sub> = ''A''1<sub>{''t'' > ''T''}</sub> for stopping times ''T'' and ''F''<sub>''T''</sub> -measurable random variables ''A''. The integral ''H'' · ''X'' for any such simple predictable process ''H'' and real valued process ''X'' is
			:<math>H\cdot X_t\equiv 1_{\{t>T\}}A(X_t-X_T).</math>
			This is extended to all simple predictable processes by the linearity of ''H'' · ''X'' in ''H''.

			A real valued process ''X'' is a semimartingale if it is càdlàg, adapted, and for every ''t'' ≥ 0,

			:<math>\left\{H\cdot X_t:H{\rm\ is\ simple\ predictable\ and\ }\|H\|\le 1\right\}</math>

			is bounded in probability. The Bichteler-Dellacherie Theorem states that these two definitions are equivalent {{Harv\|Protter\|2004\|p=144}}.

			==Examples==
			* Adapted and continuously differentiable processes are finite variation processes, and hence are semimartingales.
			* [[Wiener process\|Brownian motion]] is a semimartingale.
			* All càdlàg [[Martingale (probability theory)\|martingales]], submartingales and supermartingales are semimartingales.
			* [[Itō calculus#Itō processes\|Itō processes]], which satisfy a stochastic differential equation of the form ''dX'' = ''σdW'' + ''μdt'' are semimartingales. Here, ''W'' is a Brownian motion and ''σ, μ'' are adapted processes.
			* Every [[Lévy process]] is a semimartingale.

			Although most continuous and adapted processes studied in the literature are semimartingales, this is not always the case.

			* [[Fractional Brownian motion]] with Hurst parameter ''H'' ≠ 1/2 is not a semimartingale.

			==Properties==

			* The semimartingales form the largest class of processes for which the [[Itō calculus\|Itō integral]] can be defined.
			* Linear combinations of semimartingales are semimartingales.
			* Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the [[Stochastic calculus\|Itō integral]].
			* The [[quadratic variation]] exists for every semimartingale.
			* The class of semimartingales is closed under [[stopped process\|optional stopping]], [[Stopping time#Localization\|localization]], [[change of time]] and absolutely continuous [[Absolutely_continuous#Absolute_continuity_of_measures\|change of measure]].
			* If ''X'' is an '''R'''<sup>''m''</sup> valued semimartingale and ''f'' is a twice continuously differentiable function from '''R'''<sup>''m''</sup> to '''R'''<sup>''n''</sup>, then ''f''(''X'') is a semimartingale. This is a consequence of [[Itō's lemma]].
			* The property of being a semimartingale is preserved under shrinking the filtration. More precisely, if ''X'' is a semimartingale with respect to the filtration ''F''<sub>t</sub>, and is adapted with respect to the subfiltration ''G''<sub>t</sub>, then ''X'' is a ''G''<sub>t</sub>-semimartingale.
			* (Jacod's Countable Expansion) The property of being a semimartingale is preserved under enlarging the filtration by a countable set of disjoint sets. Suppose that ''F''<sub>t</sub> is a filtration, and ''G''<sub>t</sub> is the filtration generated by ''F''<sub>t</sub> and a countable set of disjoint measurable sets. Then, every ''F''<sub>t</sub>-semimartingale is also a ''G''<sub>t</sub>-semimartingale. {{Harv\|Protter\|2004\|p=53}}

			==Semimartingale decompositions==
			By definition, every semimartingale is a sum of a local martingale and a finite variation process. However, this decomposition is not unique.

			===Continuous semimartingales===
			A continuous semimartingale uniquely decomposes as ''X'' = ''M'' + ''A'' where ''M'' is a continuous local martingale and ''A'' is a continuous finite variation process starting at zero. {{Harv\|Rogers\|Williams\|1987\|p=358}}

			For example, if ''X'' is an Itō process satisfying the stochastic differential equation d''X''<sub>t</sub> = σ<sub>t</sub> d''W''<sub>t</sub> + ''b''<sub>t</sub> dt, then
			:<math>M_t=X_0+\int_0^t\sigma_s\,dW_s,\ A_t=\int_0^t b_s\,ds.</math>

			===Special semimartingales===
			A special semimartingale is a real valued process ''X'' with the decomposition ''X'' = ''M'' + ''A'', where ''M'' is a local martingale and ''A'' is a predictable finite variation process starting at zero. If this decomposition exists, then it is unique up to a P-null set.

			Every special semimartingale is a semimartingale. Conversely, a semimartingale is a special semimartingale if and only if the process ''X''<sub>t</sub><sup>*</sup> &equiv; sup<sub>''s'' ≤ ''t''</sub> \|X<sub>''s''</sub>\| is [[Stopping time#Localization\|locally integrable]] {{Harv\|Protter\|2004\|p=130}}.

			For example, every continuous semimartingale is a special semimartingale, in which case ''M'' and ''A'' are both continuous processes.

			===Purely discontinuous semimartingales===
			A semimartingale is called purely discontinuous if its quadratic variation [''X''] is a pure jump process,
			:<math>[X]_t=\sum_{s\le t}\Delta X_s^2</math>.
			Every adapted finite variation process is a purely discontinuous semimartingale. A continuous process is a purely discontinuous semimartingale if and only if it is an adapted finite variation process.

			Then, every semimartingale has the unique decomposition ''X'' = ''M'' + ''A'' where ''M'' is a continuous local martingale and ''A'' is a purely discontinuous semimartingale starting at zero. The local martingale ''M'' - ''M''<sub>0</sub> is called the continuous martingale part of ''X'', and written as ''X''<sup>c</sup> ({{Harvnb\|He\|Wang\|Yan\|1992\|p=209}}; {{Harvnb\|Kallenberg\|2002\|p=527}}).

			In particular, if ''X'' is continuous, then ''M'' and ''A'' are continuous.

			==Semimartingales on a manifold==

			The concept of semimartingales, and the associated theory of stochastic calculus, extends to processes taking values in a [[differentiable manifold]]. A process ''X'' on the manifold ''M'' is a semimartingale if ''f''(''X'') is a semimartingale for every smooth function ''f'' from ''M'' to '''R'''. {{Harv\|Rogers\|1987\|p=24}} Stochastic calculus for semimartingales on general manifolds requires the use of the [[Stratonovich integral]].

			==See also==
			*[[Sigma-martingale]]

			==References==
			*{{Citation\|last=He\|first=Sheng-wu\|last2=Wang\|first2=Jia-gang\|last3=Yan\|first3=Jia-an\|year=1992\|title=Semimartingale Theory and Stochastic Calculus\|publisher=Science Press, CRC Press Inc.\|isbn= 0-8493-7715-3}}
			*{{Citation\|last=Kallenberg\|first=Olav\|year=2002\|title=Foundations of Modern Probability\|edition=2nd\|publisher=Springer\|isbn=0-387-95313-2}}
			*{{Citation\|last=Protter\|first=Philip E.\|year=2004\|title=Stochastic Integration and Differential Equations\|publisher=Springer\|edition=2nd\|isbn=3-540-00313-4}}
			*{{Citation\|last=Rogers\|first=L.C.G.\|last2=Williams\|first2=David\|year=1987\|volume=2\|title=Diffusions, Markov Processes, and Martingales\|publisher=John Wiley & Sons Ltd\|isbn=0-471-91482-7}}

			{{Stochastic processes}}

			[[Category:Stochastic processes]]
			[[Category:Martingale theory]]

en>RedBot: r2.7.2) (Robot: Adding ca:Posinomi

2012-07-10T06:28:13Z

r2.7.2) (Robot: Adding ca:Posinomi

New page

Claude is her title and she totally digs that title. The factor I adore most bottle tops gathering and now I have time to take on new issues. For years she's been living in Kansas. I am a manufacturing and distribution officer.<br><br>my page; [http://pchelpnow.biz/ActivityFeed/MyProfile/tabid/60/userId/48445/Default.aspx pchelpnow.biz]