Level set (data structures)

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

• $(\mathrm{\Omega },\mathcal{ℱ},\mathbb{\mathbb{P}})$ be a probability space;
• $(\mathbb{𝕏},\mathcal{𝒜})$ be a measurable space, the state space;
• $\{{\mathcal{ℱ}}_t|t\ge 0\}$ be a filtration of the sigma algebra $\mathcal{ℱ}$;
• $X:[0,\mathrm{\infty })\times \mathrm{\Omega }\to \mathbb{𝕏}$ be a stochastic process (the index set could be $[0,T]$ or ${\mathbb{\mathbb{N}}}_0$ instead of $[0,\mathrm{\infty })$).

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

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

Properties

• It can be shown that $L^2(B)$, the space of stochastic processes $X:[0,T]\times \mathrm{\Omega }\to {\mathbb{ℝ}}^n$ for which the Ito integral

$\underset{0}{\overset{T}{\int }}{X}_t\mathrm{d}{B}_t$

with respect to Brownian motion $B$ is defined, is the set of equivalence classes of $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{g}$-measurable processes in $L^2([0,T]\times \mathrm{\Omega };{\mathbb{ℝ}}^n)$.

• 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.

References

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