Progressively measurable process

Quality 0.50 · 1 view · Updated 2 months ago

title: "Progressively measurable process" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["stochastic-processes", "measure-theory"] topic_path: "general/stochastic-processes" source: "https://en.wikipedia.org/wiki/Progressively_measurable_process" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. Progressively measurable processes are important in the theory of Itô integrals.

Definition

Let

(\Omega, \mathcal{F}, \mathbb{P}) be a probability space;
(\mathbb{X}, \mathcal{A}) be a measurable space, the state space;
{ \mathcal{F}_{t} \mid t \geq 0 } be a filtration of the sigma algebra \mathcal{F};
X : [0, \infty) \times \Omega \to \mathbb{X} be a stochastic process (the index set could be [0, T] or \mathbb{N}_{0} instead of [0, \infty));
\mathrm{Borel}([0, t]) be the Borel sigma algebra on [0,t].

The process X is said to be progressively measurable (or simply progressive) if, for every time t, the map [0, t] \times \Omega \to \mathbb{X} defined by (s, \omega) \mapsto X_{s} (\omega) is \mathrm{Borel}([0, t]) \otimes \mathcal{F}{t}-measurable. This implies that X is \mathcal{F}{t} -adapted.

A subset P \subseteq [0, \infty) \times \Omega is said to be progressively measurable if the process X_{s} (\omega) := \chi_{P} (s, \omega) is progressively measurable in the sense defined above, where \chi_{P} is the indicator function of P. The set of all such subsets P form a sigma algebra on [0, \infty) \times \Omega, denoted by \mathrm{Prog}, and a process X is progressively measurable in the sense of the previous paragraph if, and only if, it is \mathrm{Prog}-measurable.

Properties

It can be shown that L^2 (B), the space of stochastic processes X : [0, T] \times \Omega \to \mathbb{R}^n for which the Itô integral :: \int_0^T X_t , \mathrm{d} B_t : with respect to Brownian motion B is defined, is the set of equivalence classes of \mathrm{Prog}-measurable processes in L^2 ([0, T] \times \Omega; \mathbb{R}^n).
Every adapted process with left- or right-continuous paths is progressively measurable. Consequently, every adapted process with càdlàg paths is progressively measurable.
Every measurable and adapted process has a progressively measurable modification.

References

(1991). "Brownian Motion and Stochastic Calculus". Springer.
Pascucci, Andrea. (2011). "PDE and Martingale Methods in Option Pricing". Springer.

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::