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2013-09-04T21:30:23Z

99.100.102.135: /* Tool Life Expectancy */

{{Unreferenced|date=December 2009}}
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.

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''•''X'' = (''F''''i''''X'')''i''∈''I'' given by

:<math>F_{i}^{X} = \sigma \left\{ \left. X_{j}^{-1} (A) \right| j \in I, j \leq i, A \in \Sigma \right\},</math>

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''.

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'''''n''.

Any stochastic process ''X'' is an [[adapted process]] with respect to its natural filtration.

==See also==
* [[Filtration (mathematics)]]

{{DEFAULTSORT:Natural Filtration}}
[[Category:Stochastic processes]]

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