Progressively measurable process explained

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.^[1] Progressively measurable processes are important in the theory of Itô integrals.

Definition

Let

(\Omega, l{F}, P ) be a probability space;( X , l{A}) be a measurable space, the state space;\{ l{F} _{t} \mid t \geq 0 \} be a filtration of the sigma algebra l{F} ;X : [ 0 , inf ty) x \Omega \to X be a stochastic process (the index set could be [ 0 , T] or N _{0} instead of [ 0 , inf ty) );Borel ([ 0 , t]) be the Borel sigma algebra on [ 0 ,t] .

The process

X is said to be progressively measurable^[2] (or simply progressive) if, for every time t , the map [ 0 , t] x \Omega \to X defined by (s, \omega) \mapsto X _{s} (\omega) is Borel ([ 0 , t]) ⊗ l{F} _{t} -measurable. This implies that X is l{F} _{t} -adapted.

A subset

P \subseteq [ 0 , inf ty) x \Omega is said to be progressively measurable if the process X _{s} (\omega) : = \chi _{P} (s, \omega) is progressively measurable in the sense defined above, where \chi _{P} is the indicator function of P . The set of all such subsets P form a sigma algebra on [ 0 , inf ty) x \Omega , denoted by Prog , and a process X is progressively measurable in the sense of the previous paragraph if, and only if, it is Prog -measurable.

Properties

• It can be shown that

L ^{2} (B) , the space of stochastic processes X : [ 0 , T] x \Omega \to R ^{n} for which the Itô integral\int ^{T} _{0} X _{t} d B _{t} B is defined, is the set of equivalence classes of Prog -measurable processes in L ^{2} ([ 0 , T] x \Omega; 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.

Notes and References

1. Book: Karatzas. Ioannis. Shreve. Steven. 1991. Brownian Motion and Stochastic Calculus. Springer. 2nd. 0-387-97655-8. 4–5.
2. Book: Pascucci, Andrea. PDE and Martingale Methods in Option Pricing. 2011. Springer. 978-88-470-1780-1. 110. Continuous-time stochastic processes. Bocconi & Springer Series . 10.1007/978-88-470-1781-8. 118113178 .