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
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|
|Date of creation||2013-03-22 18:37:13|
|Last modified on||2013-03-22 18:37:13|
|Last modified by||gel (22282)|
|Synonym||filtration of sigma-algebras|