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| {{Unreferenced|date=December 2009}}
| | The author is known by the name of Numbers Lint. To collect cash is what her family members and her appreciate. North Dakota is her birth place but she will have to move one day or an additional. Since she was 18 she's been operating as a meter reader but she's usually wanted her own business.<br><br>Have a look at my site std testing at home ([http://i4p.info/article.php?id=114136 simply click the next document]) |
| In the theory of [[stochastic processes]] in [[mathematics]] and [[statistics]], the '''natural filtration''' associated to a stochastic process is a [[filtration (abstract algebra)|filtration]] associated to the process which records its "past behaviour" at each time. It is in a sense the simplest filtration available for studying the given process: all information concerning the process, and only that information, is available in the natural filtration.
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| More formally, let (Ω, ''F'', '''P''') be a [[probability space]]; let (''I'', ≤) be a [[total order|totally ordered]] [[index set]]; let (''S'', Σ) be a [[measurable space]]; let ''X'' : ''I'' × Ω → ''S'' be a stochastic process. Then the '''natural filtration of''' ''F'' '''with respect to''' ''X'' is defined to be the filtration ''F''<sub>•</sub><sup>''X''</sup> = (''F''<sub>''i''</sub><sup>''X''</sup>)<sub>''i''∈''I''</sub> given by
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| :<math>F_{i}^{X} = \sigma \left\{ \left. X_{j}^{-1} (A) \right| j \in I, j \leq i, A \in \Sigma \right\},</math>
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| i.e., the smallest [[sigma algebra|''σ''-algebra]] on Ω that contains all pre-images of Σ-measurable subsets of ''S'' for "times" ''j'' up to ''i''.
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| In many examples, the index set ''I'' is the [[natural numbers]] '''N''' (possibly including 0) or an [[interval (mathematics)|interval]] [0, ''T''] or [0, +∞); the state space ''S'' is often the [[real line]] '''R''' or [[Euclidean space]] '''R'''<sup>''n''</sup>.
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| Any stochastic process ''X'' is an [[adapted process]] with respect to its natural filtration.
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| ==See also==
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| * [[Filtration (mathematics)]]
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| {{DEFAULTSORT:Natural Filtration}}
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| [[Category:Stochastic processes]]
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Latest revision as of 08:54, 3 January 2015
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