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In mathematics, progressive measurability is a property of stochastic processes. A progressively measurable process cannot "see into the future", but being progressively measurable is a strictly stronger property than the notion of being an adapted process.
[edit] Definition
Let
The process X is said to be progressively measurable (or simply progressive) if, for every time t, the map ![[0, t] \times \Omega \to \mathbb{X}](http://upload.wikimedia.org/math/3/2/3/323e2c2893f050592446f7fa86c8122c.png)
defined by 
is ![\mathrm{Borel}([0, t]) \otimes \mathcal{F}_{t}](http://upload.wikimedia.org/math/d/e/e/deee56d4ba6263261cca75865935a931.png)
-measurable. This implies that X is 
-adapted.
Also, we say that a subset 
is progressively measurable if the process Xs(ω): = χP(s,ω) is progressively measurable in the sense defined above. The set of all such subsets P form a sigma algebra on 
, denoted Prog, and a process X is progressively measurable in the sense of the previous paragraph if, and only if, it is Prog-measurable.
[edit] Properties
- It can be shown that L2(B), the space of stochastic processes
![X : [0, T] \times \Omega \to \mathbb{R}^{n}](http://upload.wikimedia.org/math/5/e/a/5eac274cdacec2c5263c68d0c4bd797b.png)
for which the [[Ito integral
- with respect to Brownian motion B is defined, is the set of equivalence classes of Prog-measurable processes in
![L^2 ([0, T] \times \Omega; \mathbb{R}^n)\,](http://upload.wikimedia.org/math/0/8/0/0803a5ffe512b48b4ac983f2cf413fe7.png)
.
- Any adapted process with left- or right-continuous paths is progressively measurable.
- Consequently, any adapted process with càdlàg paths is progressively measurable.
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