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 Itō integrals.
- 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 ).
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.
- 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.
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