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

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

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

is called a filtration, if ${\displaystyle {\mathcal {F}}_{k}\subseteq {\mathcal {F}}_{\ell }}${\displaystyle {\mathcal {F}}_{k}\subseteq {\mathcal {F}}_{\ell }} for all ${\displaystyle k\leq \ell }${\displaystyle k\leq \ell }. So filtrations are families of σ-algebras that are ordered non-decreasingly.^[1] If ${\displaystyle \mathbb {F} }${\displaystyle \mathbb {F} } is a filtration, then ${\displaystyle (\Omega ,{\mathcal {A}},\mathbb {F} ,P)}${\displaystyle (\Omega ,{\mathcal {A}},\mathbb {F} ,P)} is called a filtered probability space.

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

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

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

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

${\displaystyle \sigma (X_{k}\mid k\leq n)\subseteq \sigma (X_{k}\mid k\leq n+1).}${\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 ${\displaystyle {\mathcal {A}}}${\displaystyle {\mathcal {A}}} with respect to ${\displaystyle (X_{n})_{n\in \mathbb {N} }}${\displaystyle (X_{n})_{n\in \mathbb {N} }}.

If ${\displaystyle \mathbb {F} =({\mathcal {F}}_{i})_{i\in I}}${\displaystyle \mathbb {F} =({\mathcal {F}}_{i})_{i\in I}} is a filtration, then the corresponding right-continuous filtration is defined as^[2]

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

with

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

The filtration ${\displaystyle \mathbb {F} }${\displaystyle \mathbb {F} } itself is called right-continuous if ${\displaystyle \mathbb {F} ^{+}=\mathbb {F} }${\displaystyle \mathbb {F} ^{+}=\mathbb {F} }.^[3]

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

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

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

If a filtration is an augmented filtration, it is said to satisfy the usual hypotheses or the usual conditions.^[3]

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

• Probability theory

• Articles with short description
• Short description matches Wikidata