Progressively measurable process

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.{{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}} 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{{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 }} (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

Category:Stochastic processes

Category:Measure theory