progressively measurable process

A stochastic processMathworldPlanetmath (Xt)t+ is said to be adapted to a filtrationPlanetmathPlanetmath ( (t) on the measurable spaceMathworldPlanetmathPlanetmath (Ω,) if Xt is an t-measurable random variableMathworldPlanetmath for each t=0,1,. However, for continuous-time processes, where the time t ranges over an arbitrary index setMathworldPlanetmathPlanetmath 𝕋, the property of being adapted is too weak to be helpful in many situations. Instead, considering the process as a map


it is useful to consider the measurability of X.

The process X is progressive or progressively measurable if, for every s𝕋, the stopped process XtsXmin(s,t) is (𝕋)s-measurable. In particular, every progressively measurable process will be adapted and jointly measurable. In discrete time, when 𝕋 is countableMathworldPlanetmath, the converseMathworldPlanetmath holds and every adapted process is progressive.

A set S𝕋×Ω is said to be progressive if its characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath 1S is progressive. Equivalently,


for every s𝕋. The progressively measurable sets form a σ-algebra, and a stochastic process is progressive if and only if it is measurable with respect to this σ-algebra.

Title progressively measurable process
Canonical name ProgressivelyMeasurableProcess
Date of creation 2013-03-22 18:37:31
Last modified on 2013-03-22 18:37:31
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Definition
Classification msc 60G05
Synonym progressive process
Related topic PredictableProcess
Related topic OptionalProcess
Defines progressive
Defines progressively measurable