Progressively measurable process

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 Itô integrals.

Definition

Let

• $(\Omega ,{\mathcal {F}},\mathbb {P} )$ be a probability space;
• $(\mathbb {X} ,{\mathcal {A}})$ be a measurable space, the state space;
• $\{{\mathcal {F}}_{t}\mid t\geq 0\}$ 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 $[0,\infty )$);
• $\mathrm {Borel} ([0,t])$ be the Borel sigma algebra on $[0,t]$.

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 ${\mathcal {F}}_{t}$-adapted.

A subset $P\subseteq [0,\infty )\times \Omega$ 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 $[0,\infty )\times \Omega$, denoted by $\mathrm {Prog}$, and a process $X$ is progressively measurable in the sense of the previous paragraph if, and only if, it is $\mathrm {Prog}$-measurable.

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 Itô 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 $\mathrm {Prog}$-measurable processes in $L^{2}([0,T]\times \Omega ;\mathbb {R} ^{n})$.

• 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.