Predictable process: Difference between revisions

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== 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 [[measurable function|measurable]] with respect to the [[sigma algebra|&sigma;-algebra]] <math>\mathcal{F}_n</math> for each ''n''.<ref name="Zanten">{{cite web|title=An Introduction to Stochastic Processes in Continuous Time|first1=Harry|last1=van Zanten|date=November 8, 2004|url=http://www.cs.vu.nl/~rmeester/onderwijs/stochastic_processes/sp_new.pdf|format=pdf|accessdate=October 14, 2011}}</ref>

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== Examples ==

* Every [[deterministic system|deterministic process]] is a predictable process.{{cncitation needed|date=October 2011}}

* Every continuous-time process that is [[left continuous]] is a predictable process.{{cncitation needed|date=October 2011}}

== See also ==

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[[Category:Stochastic processes]]

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