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.
Let
( Ω , F , P ) be a probability space; ( X , A ) be a measurable space, the state space; { F t ∣ t ≥ 0 } be a filtration of the sigma algebra F ; X : [ 0 , ∞ ) × Ω → X be a stochastic process (the index set could be [ 0 , T ] or N 0 instead of [ 0 , ∞ ) ).The process X is said to be progressively measurable (or simply progressive) if, for every time t , the map [ 0 , t ] × Ω → X defined by ( s , ω ) ↦ X s ( ω ) is B o r e l ( [ 0 , t ] ) ⊗ F t -measurable. This implies that X is F t -adapted.
A subset P ⊆ [ 0 , ∞ ) × Ω is said to be progressively measurable if the process X s ( ω ) := χ P ( s , ω ) is progressively measurable in the sense defined above, where χ P is the indicator function of P . The set of all such subsets P form a sigma algebra on [ 0 , ∞ ) × Ω , denoted by P r o g , and a process X is progressively measurable in the sense of the previous paragraph if, and only if, it is P r o g -measurable.
It can be shown that L 2 ( B ) , the space of stochastic processes X : [ 0 , T ] × Ω → R n for which the Ito integralwith respect to
Brownian motion B is defined, is the set of
equivalence classes of
P r o g -measurable processes in
L 2 ( [ 0 , T ] × Ω ; R n ) .
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.