# Adapted process

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{{Short description|Stochastic process}}
In the study of [stochastic processes](/source/stochastic_processes), a stochastic process is '''adapted''' (also referred to as a '''non-anticipating''' or '''non-anticipative process''') if information about the value of the process at a given time is available at that same time. An informal interpretation<ref>{{cite book|last=Wiliams|first=David|year=1979|title=Diffusions, Markov Processes and Martingales: Foundations|volume=1|publisher=Wiley|isbn=0-471-99705-6|section=II.25}}</ref> is that ''X'' is adapted if and only if, for every realisation and every ''n'', ''X<sub>n</sub>'' is known at time ''n''. The concept of an adapted process is essential, for instance, in the definition of the [Itō integral](/source/It%C5%8D_integral), which only makes sense if the [integrand](/source/integrand) is an adapted process.

==Definition==
Let
* <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [probability space](/source/probability_space);
* <math>I</math> be an index set with a total order <math>\leq</math> (often, <math>I</math> is <math>\mathbb{N}</math>, <math>\mathbb{N}_0</math>, <math>[0, T]</math> or <math>[0, +\infty)</math>);
* <math>\left(\mathcal{F}_i\right)_{i \in I}</math> be a [filtration](/source/Filtration_(probability_theory)) of the [sigma algebra](/source/sigma_algebra) <math>\mathcal{F}</math>;
* <math>(S,\Sigma)</math> be a [measurable space](/source/measurable_space), the ''state space'';
* <math>X_i: I \times \Omega \to S</math> be a [stochastic process](/source/stochastic_process).

The stochastic process <math>(X_i)_{i\in I}</math> is said to be '''adapted to the filtration''' <math>\left(\mathcal{F}_i\right)_{i \in I}</math> if the [random variable](/source/random_variable) <math>X_i: \Omega \to S</math> is a <math>(\mathcal{F}_i, \Sigma)</math>-[measurable function](/source/measurable_function) for each <math>i \in I</math>.<ref>{{cite book|last=Øksendal|first=Bernt|year=2003|title=Stochastic Differential Equations|page=25|isbn=978-3-540-04758-2|publisher=Springer}}</ref>

==Examples==
Consider a stochastic process ''X'' : [0, ''T''] × Ω → '''R''', and equip the [real line](/source/real_line) '''R''' with its usual [Borel sigma algebra](/source/Borel_sigma_algebra) generated by the [open sets](/source/open_sets).

* If we take the [natural filtration](/source/natural_filtration) ''F''<sub>•</sub><sup>''X''</sup>, where ''F''<sub>''t''</sub><sup>''X''</sup> is the ''σ''-algebra generated by the pre-images {{nowrap|''X''<sub>''s''</sub><sup>−1</sup>(''B'')}} for Borel subsets ''B'' of '''R''' and times 0 ≤ ''s'' ≤ ''t'', then ''X'' is automatically ''F''<sub>•</sub><sup>''X''</sup>-adapted. Intuitively, the natural filtration ''F''<sub>•</sub><sup>''X''</sup> contains "total information" about the behaviour of ''X'' up to time&nbsp;''t''.
* This offers a simple example of a non-adapted process {{nowrap|''X'' : [0, 2] × Ω → '''R'''}}: set ''F''<sub>''t''</sub> to be the trivial ''σ''-algebra {∅, Ω} for times 0&nbsp;≤&nbsp;''t''&nbsp;<&nbsp;1, and ''F''<sub>''t''</sub> = ''F''<sub>''t''</sub><sup>''X''</sup> for times {{nowrap|1 ≤ ''t'' ≤ 2}}. Since the only way that a function can be measurable with respect to the trivial ''σ''-algebra is to be constant, any process ''X'' that is non-constant on [0, 1] will fail to be ''F''<sub>•</sub>-adapted. The non-constant nature of such a process "uses information" from the more refined "future" ''σ''-algebras ''F''<sub>''t''</sub>, {{nowrap|1 ≤ ''t'' ≤ 2}}.

==See also==
* [Predictable process](/source/Predictable_process)
* [Progressively measurable process](/source/Progressively_measurable_process)

==References==
{{Reflist}}

Category:Stochastic processes

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Adapted from the Wikipedia article [Adapted process](https://en.wikipedia.org/wiki/Adapted_process) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Adapted_process?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.