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 The process X is said to be ****progressively measurable (or simply progressive) if, for every time t, the map defined by is -measurable. This implies that X is -adapted. A subset is said to be progressively measurable if the process 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, denoted by , and a process X is progressively measurable in the sense of the previous paragraph if, and only if, it is -measurable.

Properties