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.
Definition
Let
be a
probability space and let
be an
index set with a
total order (often
,
, or a subset of
).
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
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
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
:
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
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null s ...
.
A filtration
is called a complete filtration, if every
contains
. This implies
is a
complete measure space
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (''X'', Σ, ''μ'') is c ...
for every
(The converse is not necessarily true.)
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.
See also
*
Natural filtration
*
Filtration (mathematics)
*
Filter (mathematics)
References
[ ]
[ ]
[{{cite book , last1=Klenke , first1=Achim , year=2008 , title=Probability Theory , url=https://archive.org/details/probabilitytheor00klen_646 , url-access=limited , location=Berlin , publisher=Springer , doi=10.1007/978-1-84800-048-3 , isbn=978-1-84800-047-6, pag]
462
Probability theory