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.
Let denote the σ-algebra generated by the random variables.
Then
is a σ-algebra and is a filtration.
really is a filtration, since by definition all are σ-algebras and
This is known as the natural filtration of with respect to.

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 a probability space, and 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 implies is 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 refining.
If a filtration is an augmented filtration, it is said to satisfy the usual hypotheses or the usual conditions.