Predictable process: Difference between revisions

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In [[stochastic analysis]], a part of the mathematical theory of [[probability]], a '''predictable process''' is a [[stochastic process]] which thewhose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all [[Adapted process|adapted]] left-continuous processes.{{clarify|date=October 2011}}

== 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|&sigma;-algebra]] <math>\mathcal{F}_n</math> for each ''n''.<ref name="Zanten">{{cite web|title=An Introduction to Stochastic Processes in Continuous Time|author=Harry 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>

=== Continuous Discrete-time process ===

Given a [[filtered probability space]] <math>(\Omega,\mathcal{F},(\mathcal{F}_t_n)_{tn \geqin 0\mathbb{N}},\mathbb{P})</math>, then a stochastic process <math>(X_tX_n)_{tn \geqin 0\mathbb{N}}</math> is ''predictable'' if <math>X_{tn+1}</math> is measureable[[measurable function|measurable]] with respect to the [[sigma algebra|&sigma;-algebra]] <math>\mathcal{F}_{t^-}_n</math> for each time ''tn''.<ref name="Zanten">{{citationcite neededweb|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|access-date=October 14, 2011 |archive-url=https://web.archive.org/web/20120406084950/http://www.cs.vu.nl/~rmeester/onderwijs/stochastic_processes/sp_new.pdf |archive-date=April 6, 2012 |url-status=dead}}</ref>

=== Discrete Continuous-time process ===

== Properties ==

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</math>, considered as a mapping from <math>\Omega \times \mathbb{R}_{+} </math>, is measurable with respect to the &sigma;-algebra generated by all left-continuous adapted processes.<ref>{{cite web|title=Predictable processes: properties |url=http://www.math.ku.dk/~jesper/teaching/b108/slides38.pdf |format=pdf |access-date=October 15, 2011 |url-status=dead |archive-url=https://web.archive.org/web/20120331074812/http://www.math.ku.dk/~jesper/teaching/b108/slides38.pdf |archive-date=March 31, 2012 }}</ref>

* A continuous time process which is [[left continuous]] is always a predictable process.

This [[σ-algebra]] is also called the '''predictable σ-algebra'''.

== Examples ==

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

* Every continuous-time adapted process that is [[left continuous]] is a predictable process.{{Citation needed|reason=A Wiener process has continuous paths and is not predictable.|date=May 2020}}

== See also ==

* [[Adapted process]]

* [[Martingale (probability theory)|Martingale]]

== References ==

{{Reflist}}

{{Stochastic processes}}

[[Category:Stochastic processes]]