{{Short description|Property in the mathematical theory of stochastic processes}} 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.<ref name="Karatzas">{{cite book|last1=Karatzas|first1=Ioannis|last2=Shreve|first2=Steven|year=1991|title=Brownian Motion and Stochastic Calculus|publisher=Springer|edition=2nd|isbn=0-387-97655-8|pages=4–5}}</ref> Progressively measurable processes are important in the theory of Itô integrals.

==Definition== Let * <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a probability space; * <math>(\mathbb{X}, \mathcal{A})</math> be a measurable space, the ''state space''; * <math>\{ \mathcal{F}_{t} \mid t \geq 0 \}</math> be a filtration of the sigma algebra <math>\mathcal{F}</math>; * <math>X : [0, \infty) \times \Omega \to \mathbb{X}</math> be a stochastic process (the index set could be <math>[0, T]</math> or <math>\mathbb{N}_{0}</math> instead of <math>[0, \infty)</math>); * <math>\mathrm{Borel}([0, t])</math> be the Borel sigma algebra on <math>[0,t]</math>.

The process <math>X</math> is said to be '''progressively measurable'''<ref name="Pasc">{{cite book|last=Pascucci|first=Andrea|title=PDE and Martingale Methods in Option Pricing|date=2011|publisher=Springer|isbn=978-88-470-1780-1|page=110|chapter=Continuous-time stochastic processes|series=Bocconi & Springer Series |doi=10.1007/978-88-470-1781-8|s2cid=118113178 }}</ref> (or simply '''progressive''') if, for every time <math>t</math>, the map <math>[0, t] \times \Omega \to \mathbb{X}</math> defined by <math>(s, \omega) \mapsto X_{s} (\omega)</math> is <math>\mathrm{Borel}([0, t]) \otimes \mathcal{F}_{t}</math>-measurable. This implies that <math>X</math> is <math> \mathcal{F}_{t} </math>-adapted.<ref name="Karatzas" />

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

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

==References== {{reflist}}

{{Stochastic processes}}

Category:Stochastic processes Category:Measure theory