filtration of -algebras
For an ordered set , a filtration of -algebras (http://planetmath.org/SigmaAlgebra) is a collection of -algebras on an underlying set , satisfying for all in . Here, is understood as the time variable, taking values in the index set , and represents the collection of all events observable up until time . The index set is usually a subset of the real numbers, with common examples being for discrete-time and for continuous-time scenarios. The collection is a filtration on a measurable space if for every . If, furthermore, there is a probability measure defined on the underlying measurable space then this gives a filtered probability space. The alternative notation is often used for the filtration or, when the index set is clear from the context, simply or .
Filtrations are widely used for studying stochastic processes, where a process with time ranging over the set is said to be adapted to the filtration if is an -measurable random variable for each time .
Conversely, any stochastic process generates a filtration. Let be the smallest -algebra with respect to which is measurable for all ,
This defines the smallest filtration to which is adapted, known as the natural filtration of .
Given a filtration, there are various limiting -algebras which can be defined. The values at plus and minus infinity are
which satisfy . 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 is the maximum of it is often convenient to set or, if is the minimum, . It is easily verified that for all times . Furthermore, and are themselves filtrations.
A filtration is said to be right-continuous if for every so, in particular, is always the smallest right-continuous filtration larger than .
Title | filtration of -algebras |
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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 |