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

Let

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

The process ${\displaystyle X}$ is said to be progressively measurable^[2] (or simply progressive) if, for every time ${\displaystyle t}$, the map ${\displaystyle [0,t]\times \Omega \to \mathbb {X} }$ defined by ${\displaystyle (s,\omega )\mapsto X_{s}(\omega )}$ is ${\displaystyle \mathrm {Borel} ([0,t])\otimes {\mathcal {F}}_{t}}$-measurable. This implies that ${\displaystyle X}$ is ${\displaystyle {\mathcal {F}}_{t}}$-adapted.^[1]

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

Properties

• It can be shown^[1] that ${\displaystyle L^{2}(B)}$, the space of stochastic processes ${\displaystyle X:[0,T]\times \Omega \to \mathbb {R} ^{n}}$ for which the Itô integral

${\displaystyle \int _{0}^{T}X_{t}\,\mathrm {d} B_{t}}$

with respect to Brownian motion ${\displaystyle B}$ is defined, is the set of equivalence classes of ${\displaystyle \mathrm {Prog} }$-measurable processes in ${\displaystyle L^{2}([0,T]\times \Omega$ ;\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

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