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 (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
 * $$ \mathbb F:= (\mathcal F_i)_{i \in I} $$

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 $$ (X_n)_{n \in \N} $$ 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 $$ \mathbb F= (\mathcal F_n)_{n \in \N} $$ 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 $$(X_n)_{n \in \N}$$.

Right-continuous filtration
If $$ \mathbb F= (\mathcal F_i)_{i \in I} $$ is a filtration, then the corresponding right-continuous filtration is defined as
 * $$ \mathbb F^+:= (\mathcal F_i^+)_{i \in I}, $$

with
 * $$ \mathcal F_i^+:= \bigcap_{i < z} \mathcal F_z. $$

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,
 * $$ \mathcal N_P:= \{A \subseteq \Omega \mid A \subseteq B \text{ for some } B \in \mathcal F \text{ with } P(B)=0 \} $$

be the set of all sets that are contained within a $$ P $$-null set.

A filtration $$ \mathbb F= (\mathcal F_i)_{i \in I} $$ 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 $$ \tilde {\mathbb F} $$ refining $$ \mathbb F $$.

If a filtration is an augmented filtration, it is said to satisfy the usual hypotheses or the usual conditions.