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
 * $$(\Omega, \mathcal{F}, \mathbb{P})$$ 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 (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.

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 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.
 * Every measurable and adapted process has a progressively measurable modification.