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 Itō 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 $[0,\infty )$).

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 Ito 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

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) PDE and Martingale Methods in Option Pricing. Berlin: Springer