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Predictable process

Stochastic process

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated December 15, 2025

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.

01Mathematical 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.

02Examples

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Sources and credits

This article is adapted from the Wikipedia article “Predictable process”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Adapted process

In the study of stochastic processes, a stochastic process is adapted if information about the value of the process at a given time is available at that same time. An informal interpretation is that X is adapted if and only if, for every realisation and every n, Xn is known at time n.

Martingale (probability theory)

In probability theory, a martingale is a stochastic process in which the expected value of the next observation, given all prior observations, is equal to the most recent value. In other words, the conditional expectation of the next value, given the past, is equal to the present value.