Predictable process
Stochastic process
title: "Predictable process" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["stochastic-processes"] description: "Stochastic process" topic_path: "general/stochastic-processes" source: "https://en.wikipedia.org/wiki/Predictable_process" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Stochastic 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.
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.
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. This σ-algebra is also called the predictable σ-algebra.
Examples
- Every deterministic process is a predictable process.
- Every continuous-time adapted process that is left continuous is a predictable process.
References
References
- (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time".
- "Predictable processes: properties".
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