Model of information available at a given point of a random process
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
![{\displaystyle \mathbb {F} :=({\mathcal {F}}_{i})_{i\in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de40f60a52bf51c3d5747938dfb264f371ffba29)
is called a filtration, if
for all
. So filtrations are families of σ-algebras that are ordered non-decreasingly.[1] 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
![{\displaystyle {\mathcal {F}}_{n}:=\sigma (X_{k}\mid k\leq n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2628a66c18488e4ee461914e0787394fa194e1ce)
is a σ-algebra and
is a filtration.
really is a filtration, since by definition all
are σ-algebras and
![{\displaystyle \sigma (X_{k}\mid k\leq n)\subseteq \sigma (X_{k}\mid k\leq n+1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3e9740b7967152c0eba84b7782dbeead30be9e)
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[2]
![{\displaystyle \mathbb {F} ^{+}:=({\mathcal {F}}_{i}^{+})_{i\in I},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b441fcde271e59b7f97fc8321c802f388e64c5)
with
![{\displaystyle {\mathcal {F}}_{i}^{+}:=\bigcap _{i<z}{\mathcal {F}}_{z}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/700c36b803ffd6ea8a8878eed8a87a3fe872e115)
The filtration
itself is called right-continuous if
.[3]
Complete filtration
Let
be a probability space and let,
![{\displaystyle {\mathcal {N}}_{P}:=\{A\subseteq \Omega \mid A\subseteq B{\text{ for some }}B\in {\mathcal {F}}{\text{ with }}P(B)=0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa9c49b6c931cd49b837509d439cc9b384daffb)
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
(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.[3]
See also
References