Filtered Probability Space

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 $(\Omega, \mathcal A, P)$ be a probability space and let $I$ be an index set with a total order $\leq$ (often $\N$, $\R^+$, or a subset of $\mathbb R^+$).

For every $i \in I$ let $\mathcal F_i$ be a sub-''σ''-algebra of $\mathcal A$. Then
:$\mathbb F:= (\mathcal F_i)_$

is called a filtration, if $\mathcal F_k \subseteq \mathcal F_\ell$ for all $k \leq \ell$. So filtrations are families of ''σ''-algebras that are ordered non-decreasingly. If $\mathbb F$ is a filtration, then $(\Omega, \mathcal A, \mathbb F, P)$ is called a filtered probability space.

Example

Let $(X_n)_$ be a stochastic process on the probability space $(\Omega, \mathcal A, P)$.
Let $\sigma(X_k \mid k \leq n)$ denote the ''σ''-algebra generated by the random variables $X_1, X_2, \dots, X_n$.
Then
:$\mathcal F_n:=\sigma(X_k \mid k \leq n)$

is a ''σ''-algebra and $\mathbb F= (\mathcal F_n)_$ is a filtration.

$\mathbb F$ really is a filtration, since by definition all $\mathcal F_n$ are ''σ''-algebras and
:$\sigma(X_k \mid k \leq n) \subseteq \sigma(X_k \mid k \leq n+1).$
This is known as the natural filtration of $\mathcal A$ with respect to $(X_n)_$.

Types of filtrations

Right-continuous filtration

If $\mathbb F= (\mathcal F_i)_$ is a filtration, then the corresponding right-continuous filtration is defined as
:$\mathbb F^+:= (\mathcal F_i^+)_,$

with
:$\mathcal F_i^+:= \bigcap_ \mathcal F_z.$
The filtration $\mathbb F$ itself is called right-continuous if $\mathbb F^+ = \mathbb F$.

Complete filtration

Let $(\Omega, \mathcal F, P)$ be a probability space, and let
:$\mathcal N_P:= \$
be the set of all sets that are contained within a $P$-null set.

A filtration $\mathbb F= (\mathcal F_i)_$ is called a complete filtration, if every $\mathcal F_i$ contains $\mathcal N_P$. This implies $(\Omega, \mathcal F_i, P)$ is a complete measure space for every $i \in I.$ (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 $\mathbb F$ there exists a smallest augmented filtration $\tilde$ refining $\mathbb F$.

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