filtration of σ-algebras

For an ordered set T, a filtrationPlanetmathPlanetmath of σ-algebras ( (t)tT is a collectionMathworldPlanetmath of σ-algebras on an underlying set Ω, satisfying st for all s<t in T. Here, t is understood as the time variable, taking values in the index setMathworldPlanetmathPlanetmath T, and t represents the collection of all events observable up until time t. The index set is usually a subset of the real numbers, with common examples being T=+ for discrete-time and T=+ for continuous-time scenarios. The collection (t)tT is a filtration on a measurable spaceMathworldPlanetmathPlanetmath (Ω,) if t for every t. If, furthermore, there is a probability measureMathworldPlanetmath defined on the underlying measurable space then this gives a filtered probability space. The alternative notation (t,tT) is often used for the filtration or, when the index set T is clear from the context, simply (t) or 𝐅.

Filtrations are widely used for studying stochastic processesMathworldPlanetmath, where a process Xt with time ranging over the set T is said to be adapted to the filtration if Xt is an t-measurable random variableMathworldPlanetmath for each time t.

Conversely, any stochastic process (Xt)tT generates a filtration. Let t be the smallest σ-algebra with respect to which Xs is measurable for all st,


This defines the smallest filtration to which X is adapted, known as the natural filtration of X.

Given a filtration, there are various limiting σ-algebras which can be defined. The values at plus and minus infinityMathworldPlanetmathPlanetmath are


which satisfy -t. In continuous-time, when the index set is an interval of the real numbers, the left and right limits can be defined at any time. They are,


except if t is the maximum of T it is often convenient to set t+=t or, if t is the minimum, t-=t. It is easily verified that ss+t-t for all times s<t. Furthermore, (t+) and (t-) are themselves filtrations.

A filtration is said to be right-continuous if t=t+ for every t so, in particular, (t+) is always the smallest right-continuous filtration larger than (t).

Title filtration of σ-algebras
Canonical name FiltrationOfsigmaalgebras
Date of creation 2013-03-22 18:37:13
Last modified on 2013-03-22 18:37:13
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Definition
Classification msc 60G05
Synonym filtration of sigma-algebras
Related topic FilteredProbabilitySpace
Related topic Filtration
Defines natural filtration