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. Progressively measurable processes are important in the theory of Itô integrals.

Definition

Let
* $(\Omega, \mathcal, \mathbb)$ be a probability space;
* $(\mathbb, \mathcal)$ be a measurable space, the ''state space'';
* $\$ be a filtration of the sigma algebra $\mathcal$;
* X : , \infty) \times \Omega \to \mathbb be a stochastic process (the index set could be $[0, T]$ or $\mathbb_$ instead of $[0, \infty)$);
* \mathrm( , t be the Borel sigma algebra on $[0,t]$.

The process $X$ is said to be progressively measurable (or simply progressive) if, for every time $t$, the map , t\times \Omega \to \mathbb defined by $(s, \omega) \mapsto X_ (\omega)$ is \mathrm( , t \otimes \mathcal_-measurable. This implies that $X$ is $\mathcal_$-adapted.

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

Properties

* It can be shown that $L^2 (B)$, the space of stochastic processes X : , T\times \Omega \to \mathbb^n for which the Itô integral
:: $\int_0^T X_t \, \mathrm B_t$
: with respect to Brownian motion $B$ is defined, is the set of equivalence classes of $\mathrm$-measurable processes in L^2 ( , T\times \Omega; \mathbb^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

Stochastic processes
Measure theory

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