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.

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

For every ${\displaystyle i\in I}$let ${\displaystyle {\mathcal {F}}_{i}}$be a sub-σ-algebra of ${\displaystyle {\mathcal {A}}}$. Then

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

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

Let ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$be a stochastic process on the probability space ${\displaystyle (\Omega ,{\mathcal {A}},P)}$. Let ${\displaystyle \sigma (X_{k}\mid k\leq n)}$denote the σ-algebra generated by the random variables ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$. Then

${\displaystyle {\mathcal {F}}_{n}:=\sigma (X_{k}\mid k\leq n)}$

is a σ-algebra and ${\displaystyle \mathbb {F} =({\mathcal {F}}_{n})_{n\in \mathbb {N} }}$is a filtration.

${\displaystyle \mathbb {F} }$really is a filtration, since by definition all ${\displaystyle {\mathcal {F}}_{n}}$are σ-algebras and

${\displaystyle \sigma (X_{k}\mid k\leq n)\subseteq \sigma (X_{k}\mid k\leq n+1).}$

This is known as the natural filtration of ${\displaystyle {\mathcal {A}}}$with respect to ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$.

If ${\displaystyle \mathbb {F} =({\mathcal {F}}_{i})_{i\in I}}$is a filtration, then the corresponding right-continuous filtration is defined as

${\displaystyle \mathbb {F} ^{+}:=({\mathcal {F}}_{i}^{+})_{i\in I},}$

with

${\displaystyle {\mathcal {F}}_{i}^{+}:=\bigcap _{z>i}{\mathcal {F}}_{z}.}$

The filtration ${\displaystyle \mathbb {F} }$itself is called right-continuous if ${\displaystyle \mathbb {F} ^{+}=\mathbb {F} }$.

Let ${\displaystyle (\Omega ,{\mathcal {F}},P)}$be a probability space, and let

${\displaystyle {\mathcal {N}}_{P}:=\{A\subseteq \Omega \mid A\subseteq B{\text{ for some }}B\in {\mathcal {F}}{\text{ with }}P(B)=0\}}$

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

A filtration ${\displaystyle \mathbb {F} =({\mathcal {F}}_{i})_{i\in I}}$is called a complete filtration, if every ${\displaystyle {\mathcal {F}}_{i}}$contains ${\displaystyle {\mathcal {N}}_{P}}$. This implies ${\displaystyle (\Omega ,{\mathcal {F}}_{i},P)}$is a complete measure space for every ${\displaystyle i\in I.}$(The converse is not necessarily true.)

A filtration is called an augmented filtration if it is complete and right continuous. For every filtration ${\displaystyle \mathbb {F} }$there exists a smallest augmented filtration ${\displaystyle {\tilde {\mathbb {F} }}}$refining ${\displaystyle \mathbb {F} }$.

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

• Natural filtration
• Filtration (mathematics)
• Filter (mathematics)