Progressively measurable process

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.

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 $mathcal\{F\}$;
* $X\; :\; [0,\; infty)\; imes\; Omega\; o\; mathbb\{X\}$ 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 $[0,\; t]\; imes\; Omega\; o\; mathbb\{X\}$ 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 $P\; subseteq\; [0,\; infty)\; imes\; Omega$ is progressively measurable if the process $X\_\{s\}\; (omega)\; :=\; chi\_\{P\}\; (s,\; omega)$ is progressively measurable in the sense defined above. The set of all such subsets $P$ form a sigma algebra on $[0,\; infty)\; imes\; Omega$, denoted $mathrm\{Prog\}$, 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 $X\; :\; [0,\; T]\; imes\; Omega\; o\; mathbb\{R\}^\{n\}$ 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 $mathrm\{Prog\}$-measurable processes in $L^\{2\}\; (\; [0,\; T]\; imes\; Omega;\; mathbb\{R\}^\{n\})$.
* Any adapted process with left- or right-continuous paths is progressively measurable.
* Consequently, any adapted process with càdlàg paths is progressively measurable.