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

be a probability space;
be a measurable space, the state space;
be a filtration of the sigma algebra ;
be a stochastic process ;
be the Borel sigma algebra on.

The process is said to be progressively measurable if, for every time, the map defined by is -measurable. This implies that is -adapted.
A subset is said to be progressively measurable if the process is progressively measurable in the sense defined above, where is the indicator function of. The set of all such subsets form a sigma algebra on, denoted by, and a process is progressively measurable in the sense of the previous paragraph if, and only if, it is -measurable.

Properties

It can be shown that, the space of stochastic processes for which the Itô integral
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.