The process is said to be progressively measurable[2] (or simply progressive) if, for every time , the map defined by is -measurable. This implies that is -adapted.[1]
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[1] that , the space of stochastic processes for which the
Every adapted process with left- or right-continuous paths is progressively measurable. Consequently, every adapted process with càdlàg paths is progressively measurable.[1]
Every measurable and adapted process has a progressively measurable modification.[1]