filtered probability space

A filtered probability space, or stochastic basis, (Ω,,(t)tT,) consists of a probability spaceMathworldPlanetmath (Ω,,) and a filtrationPlanetmathPlanetmath ( (t)tT contained in . Here, T is the time index setMathworldPlanetmathPlanetmath, and is an ordered set — usually a subset of the real numbers — such that st for all s<t in T.

Filtered probability spaces form the setting for defining and studying stochastic processesMathworldPlanetmath. A process Xt with time index t ranging over T is said to be adapted if Xt is an t-measurable random variableMathworldPlanetmath for every t.

When the index set T is an interval ( of the real numbers (i.e., continuous-time), it is often convenient to impose further conditions. In this case, the filtered probability space is said to satisfy the usual conditions or usual hypotheses if the following conditions are met.

  • The probability space (Ω,,) is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (

  • The σ-algebras t contain all the sets in of zero probability.

  • The filtration t is right-continuous. That is, for every non-maximal tT, the σ-algebra t+s>ts is equal to t.

Given any filtered probability space, it can always be enlarged by passing to the completion of the probability space, adding zero probability sets to t, and by replacing t by t+. This will then satisfy the usual conditions. In fact, for many types of processes defined on a complete probability space, their natural filtration will already be right-continuous and the usual conditions met. However, the process of completing the probability space depends on the specific probability measure and in many situations, such as the study of Markov processes, it is necessary to study many different measures on the same space. A much weaker condition which can be used is that the σ-algebras t are universally complete, which is still strong enough to apply much of the ‘heavy machinery’ of stochastic processes, such as the Doob-Meyer decomposition, sectionPlanetmathPlanetmath theorems, etc.

Title filtered probability space
Canonical name FilteredProbabilitySpace
Date of creation 2013-03-22 18:36:51
Last modified on 2013-03-22 18:36:51
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Definition
Classification msc 60G05
Related topic FiltrationOfSigmaAlgebras
Defines stochastic basis
Defines usual conditions
Defines usual hypotheses