Filtration (probability theory)


In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are totally ordered collections of subsets that are used to model the information that is available at a given point and therefore play an important role in the formalization of random processes.

Definition

Let be a probability space and let be an index set with a total order .
For every let be a Sub σ-algebra of. Then
is called a filtration, if for all. So filtrations are families of σ-algebras that are ordered non-decreasingly. If is a filtration, then is called a filtered probability space.

Example

Let be a stochastic process on the probability space. Then
is a σ-algebra and is a filtration. Here denotes the σ-algebra generated by the random variables.
really is a filtration, since by definition all are σ-algebras and

Types of filtrations

Right-continuous filtration

If is a filtration, then the corresponding right-continuous filtration is defined as
with
The filtration itself is called right-continuous if.

Complete filtration

Let
be the set of all sets that are contained within a -null set.
A filtration is called a complete filtration, if every contains. This is equivalent to being a complete measure space for every

Augmented filtration

A filtration is called an augmented filtration if it is complete and right continuous. For every filtration there exists a smallest augmented filtration of.
If a filtration is an augmented filtration, it is said to satisfy the usual hypotheses or the usual conditions.