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 $$\mathcal{F}$$;
 * $$X : [0, \infty) \times \Omega \to \mathbb{X}$$ 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 $$X_{s} (\omega) := \chi_{P} (s, \omega)$$ is progressively measurable in the sense defined above. The set of all such subsets $$P$$ form a sigma algebra on $$[0, \infty) \times \Omega$$, 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 $$X : [0, T] \times \Omega \to \mathbb{R}^{n}$$ 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 $$\mathrm{Prog}$$-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.