predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose 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.{{clarify|date=October 2011}}

Mathematical definition

= Discrete-time process =

Given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n \in \mathbb{N}},\mathbb{P})$ , then a stochastic process $(X_n)_{n \in \mathbb{N}}$ is predictable if $X_{n+1}$ is measurable with respect to the σ-algebra $\mathcal{F}_n$ for each n.{{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|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}}

= Continuous-time process =

Given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ , then a continuous-time stochastic process $(X_t)_{t \geq 0}$ is predictable if $X$ , considered as a mapping from $\Omega \times \mathbb{R}_{+}$ , is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.{{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 }}

This σ-algebra is also called the predictable σ-algebra.

Examples

Every 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}}

References

Category:Stochastic processes