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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

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

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

^ ^a ^b ^c ^d ^e 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. Springer. p. 110. doi:10.1007/978-88-470-1781-8. ISBN 978-88-470-1780-1. {{cite book}}: Unknown parameter |city= ignored (|location= suggested) (help)

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[Karatzas-1] Karatzas, Ioannis; Shreve, Steven (1991). Brownian Motion and Stochastic Calculus (2nd ed.). Springer. pp. 4–5. ISBN 0-387-97655-8.

[Pasc-2] Pascucci, Andrea (2011). "Continuous-time stochastic processes". PDE and Martingale Methods in Option Pricing. Springer. p. 110. doi:10.1007/978-88-470-1781-8. ISBN 978-88-470-1780-1. {{cite book}}: Unknown parameter |city= ignored (|location= suggested) (help)

[1]

[2]

@@ Line 9: / Line 9: @@
 * <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>Pascucci, Andrea (2011) ''PDE and Martingale Methods in Option Pricing''. Berlin: Springer {{Page needed|date=August 2011}}</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_function|measurable]]. This implies that <math>X</math> is <math> \mathcal{F}_{t} </math>-adapted.<ref name="Karatzas" />
+The process <math>X</math> is said to be '''progressively measurable'''<ref name=Pasc>{{cite book | last = Pascucci | first = Andrea | date = 2011 | title = PDE and Martingale Methods in Option Pricing | city = Berlin | publisher = Springer | page = 110 | chapter = Continuous-time stochastic processes | ISBN = 978-88-470-1780-1 | doi = 10.1007/978-88-470-1781-8}}</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_function|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.