Progressively Measurable Process

A Progressively Measurable Process is a stochastic process that is adapted and measurable.

• AKA: Progressive Process

See: Adapted Process, Filtration, Sigma Algebra.

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References

2016

• (Wikipedia, 2016) ⇒ Retrieved 2016-07-24
  • 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.

Definition

Let

• [math]\displaystyle{ (\Omega, \mathcal{F}, \mathbb{P}) }[/math] be a probability space;
• [math]\displaystyle{ (\mathbb{X}, \mathcal{A}) }[/math] be a measurable space, the state space ;
• [math]\displaystyle{ \{ \mathcal{F}_{t} \mid t \geq 0 \} }[/math] be a filtration of the sigma algebra [math]\displaystyle{ \mathcal{F} }[/math];
• [math]\displaystyle{ X : [0, \infty) \times \Omega \to \mathbb{X} }[/math] be a stochastic process (the index set could be [math]\displaystyle{ [0, T] }[/math] or [math]\displaystyle{ \mathbb{N}_{0} }[/math] instead of [math]\displaystyle{ [0, \infty) }[/math]).

The process [math]\displaystyle{ X }[/math] is said to be progressively measurable (or simply progressive) if, for every time [math]\displaystyle{ t }[/math], the map [math]\displaystyle{ [0, t] \times \Omega \to \mathbb{X} }[/math] defined by [math]\displaystyle{ (s, \omega) \mapsto X_{s} (\omega) }[/math] is [math]\displaystyle{ \mathrm{Borel}([0, t]) \otimes \mathcal{F}_{t} }[/math]-measurable. This implies that [math]\displaystyle{ X }[/math] is [math]\displaystyle{ \mathcal{F}_{t} }[/math]-adapted.

A subset [math]\displaystyle{ P \subseteq [0, \infty) \times \Omega }[/math] is said to be progressively measurable if the process [math]\displaystyle{ X_{s} (\omega) := \chi_{P} (s, \omega) }[/math] is progressively measurable in the sense defined above, where [math]\displaystyle{ \chi_{P} }[/math] is the indicator function of [math]\displaystyle{ P }[/math]. The set of all such subsets [math]\displaystyle{ P }[/math] form a sigma algebra on [math]\displaystyle{ [0, \infty) \times \Omega }[/math], denoted by [math]\displaystyle{ \mathrm{Prog} }[/math], and a process [math]\displaystyle{ X }[/math] is progressively measurable in the sense of the previous paragraph if, and only if, it is [math]\displaystyle{ \mathrm{Prog} }[/math]-measurable.

2012

• (Lelley, 2012) ⇒ Steven P. Lalley (2012) Notes on the Ito Calculus http://galton.uchicago.edu/~lalley/Courses/385/Old/ItoIntegral-2012.pdf
  • Definition 1. A stochastic process [math]\displaystyle{ \{X_t\}_{t\geq 0} }[/math]is said to be progressively measurable if for every [math]\displaystyle{ T \geq 0 }[/math] it is, when viewed as a function [math]\displaystyle{ X(t, \omega) }[/math] on the product space [math]\displaystyle{ ([0, T])\times \Omega }[/math], measurable relative to the product sigma−algebra [math]\displaystyle{ B_{[0,T]} × F_T }[/math].

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