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| In the study of [[stochastic processes]], an '''adapted process''' (or '''non-anticipating process''') is one that cannot "see into the future". 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]], which only makes sense if the [[integrand]] is an adapted process.
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| ==Definition== | |
| Let
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| * <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [[probability space]];
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| * <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>);
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| * <math>\mathcal{F}_{\cdot} = \left(\mathcal{F}_i\right)_{i \in I}</math> be a [[Filtration (abstract algebra)|filtration]] of the [[sigma algebra]] <math>\mathcal{F}</math>;
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| * <math>(S,\Sigma)</math> be a [[measurable space]], the ''state space'';
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| * <math>X: I \times \Omega \to S</math> be a [[stochastic process]].
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| The process <math>X</math> is said to be '''adapted to the filtration''' <math>\left(\mathcal{F}_i\right)_{i \in I}</math> if the [[random variable]] <math>X_i: \Omega \to S</math> is a <math>(\mathcal{F}_i, \Sigma)</math>-[[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>
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| ==Examples==
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| Consider a stochastic process ''X'' : [0, ''T''] × Ω → '''R''', and equip the [[real line]] '''R''' with its usual [[Borel sigma algebra]] generated by the [[open sets]].
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| * If we take the [[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 ''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 ''t''.
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| * This offers a simple example of a non-adapted process ''X'' : [0, 2] × Ω → '''R''': set ''F''<sub>''t''</sub> to be the trivial ''σ''-algebra {∅, Ω} for times 0 ≤ ''t'' < 1, and ''F''<sub>''t''</sub> = ''F''<sub>''t''</sub><sup>''X''</sup> for times 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>, 1 ≤ ''t'' ≤ 2.
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| ==See also==
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| * [[Predictable process]]
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| * [[Progressively measurable process]]
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| ==References==
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| {{Reflist}}
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| [[Category:Stochastic processes]]
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