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 (stochastic) processes.
be a probability space and let
be an index set with a total order
, or a subset of
be a sub-σ-algebra of
Then is called a filtration, if
ℓ
k ≤ ℓ
So filtrations are families of σ-algebras that are ordered non-decreasingly.
is called a filtered probability space.
be a stochastic process on the probability space
σ (
denote the σ-algebra generated by the random variables
Then is a σ-algebra and
really is a filtration, since by definition all
are σ-algebras and This is known as the natural filtration of
with respect to
is a filtration, then the corresponding right-continuous filtration is defined as[2] with The filtration
itself is called right-continuous if
be a probability space, and let be the set of all sets that are contained within a
-null set.
is called a complete filtration, if every
contains
This implies
is a complete measure space for every
(The converse is not necessarily true.)
A filtration is called an augmented filtration if it is complete and right continuous.
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.