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

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

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

Also, we say that a subset $P \subseteq [0, \infty) \times \Omega$ is progressively measurable if the process $X s (ω): = χ P (s,ω)$ is progressively measurable in the sense defined above. The set of all such subsets $P$ form a sigma algebra on $[0, \infty) \times \Omega$ , 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 $L 2 (B)$ , the space of stochastic processes $X : [0, T] \times \Omega \to \mathbb{R}^{n}$ for which the Ito integral $\int_{0}^{T} X_{t} \, \mathrm{d} B_{t}$ with respect to Brownian motion $B$ is defined, is the set of equivalence classes of $Prog$ -measurable processes in $L^{2} ([0, T] \times \Omega; \mathbb{R}^{n})$ .
Any adapted process with left- or right-continuous paths is progressively measurable.
Consequently, any adapted process with càdlàg paths is progressively measurable.