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In [[stochastic analysis]], a part of the mathematical theory of [[probability]], a '''predictable process''' is a [[stochastic process]] which the value is knowable at a prior time. The predictable processes form the smallest class{{clarify|reason=of what?}} that is closed under taking limits of sequences and contains all adapted left continuous processes{{clarify|reason=wikilink or explain meaning of this phrase)).
== Mathematical definition ==
=== Discrete-time process ===
Given a [[filtered probability space]] <math>(\Omega,\mathcal{F},(\mathcal{F}_n)_{n \in \mathbb{N}},\mathbb{P})</math>, then a stochastic process <math>(X_n)_{n \in \mathbb{N}}</math> is ''predictable'' if <math>X_{n+1}</math> is [[measureable function|measureable]] with respect to the [[sigma algebra|σ-algebra]] <math>\mathcal{F}_n</math> for each ''n''.<ref name="Zanten">{{cite web|title=An Introduction to Stochastic Processes in Continuous Time|
=== Continuous-time process ===
Given a filtered probability space <math>(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})</math>, then a [[continuous-time stochastic process]] <math>(X_t)_{t \geq 0}</math> is ''predictable'' if <math>X_{t}</math> is measureable with respect to the σ-algebra <math>\mathcal{F}_{t^-}</math> for each time ''t''.<ref>{{cite web|title=Predictable processes: properties|url=http://www.math.ku.dk/~jesper/teaching/b108/slides38.pdf|format=pdf|accessdate=October 15, 2011}}</ref>
== Examples ==
* Every [[deterministic system|deterministic process]] is a predictable process.{{cn|date=October 2011}}
* Every continuous-time process that is [[left continuous]] is a predictable process.{{cn|date=October 2011}}
== See also ==
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== References ==
{{Reflist}}
{{probability-stub}}▼
[[Category:Stochastic processes]]
▲{{probability-stub}}
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