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.^[1] Progressively measurable processes are important in the theory of Itô integrals.

Definition

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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^[2] (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.^[1]

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

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It can be shown^[1] 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.^[1]
Every measurable and adapted process has a progressively measurable modification.^[1]

References

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1 2 3 4 5 Karatzas, Ioannis; Shreve, Steven (1991). Brownian Motion and Stochastic Calculus (2nd ed.). Springer. pp. 4–5. ISBN 0-387-97655-8.
↑ Pascucci, Andrea (2011). "Continuous-time stochastic processes". PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series. Springer. p. 110. doi:10.1007/978-88-470-1781-8. ISBN 978-88-470-1780-1. S2CID 118113178.

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