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

• be a probability space;

• be a measurable space, the state space;

• be a filtration of the sigma algebra;

• be a stochastic process);

• be the Borel sigma algebra.

is said to be progressively measurable (or simply progressive-measurable-adapted.

is said to be progressively measurable is the indicator function-measurable.

Properties

• It can be shown for which the Itô integral

• with respect to Brownian motion is defined, is the set of equivalence classes.

• 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