Progressively measurable process
From Wikipedia, the free encyclopedia
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
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 [0,T] or 
instead of 
).
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}](../../../math/3/2/3/323e2c2893f050592446f7fa86c8122c.png)
defined by 
is ![\mathrm{Borel}([0, t]) \otimes \mathcal{F}_{t}](../../../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}](../../../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})](../../../math/3/7/1/3718902d7e4b28e786b644f3b89b043e.png)
. - Any adapted process with left- or right-continuous paths is progressively measurable.
- Consequently, any adapted process with càdlàg paths is progressively measurable.
