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The phrase "stochastic analysis" doesn't tell the lay reader that mathematics is what this is about. |
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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 that is closed under taking limits of sequences and contains all adapted left continuous processes.
== Mathematical definition ==
=== Discrete
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|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
Given a filtered probability space <math>(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})</math>, then a 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 ==
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== See also ==
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