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 (the index set could be or instead of ); be the Borel sigma algebra on .
The process is said to be progressively measurable (or simply progressive) 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
with respect to Brownian motion is defined, is the set of equivalence classes of -measurable processes in . 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