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 processes.
Definition
Let \( {\displaystyle (\Omega ,{\mathcal {A}},P)} \) be a probability space and let I be an index set with a total order \( \leq \) (often \( {\displaystyle \mathbb {N} } \), \( {\displaystyle \mathbb {R} ^{+}} \), or a subset of \({\displaystyle \mathbb {R} ^{+}} \) ).
For every \( i\in I\) 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 F \( {\displaystyle {\mathcal {F}}_{k}\subseteq {\mathcal {F}}_{\ell }\subseteq {\mathcal {A}}} \) for all \( {\displaystyle k\leq \ell } \). So filtrations are families of σ-algebras that are ordered non-decreasingly.[1] If \( {\mathbb F} \) is a filtration, then \( {\displaystyle (\Omega ,{\mathcal {A}},\mathbb {F} ,P)} \) is called a filtered probability space.
Example
Let \( {\displaystyle (X_{n})_{n\in \mathbb {N} }} \) be a stochastic process on the probability space \( {\displaystyle (\Omega ,{\mathcal {A}},P)} \) . 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. Here \( {\displaystyle \sigma (X_{k}\mid k\leq n)} \) denotes the σ-algebra generated by the random variables \( {\displaystyle X_{1},X_{2},\dots ,X_{n}} \).
F {\displaystyle \mathbb {F} } {\mathbb F} really is a filtration, since by definition all \( {\displaystyle {\mathcal {F}}_{n}} \) are σ-algebras and
\( {\displaystyle \sigma (X_{k}\mid k\leq n)\subseteq \sigma (X_{k}\mid k\leq n+1).} \)
Types of filtrations
Right-continuous filtration
If \( {\displaystyle \mathbb {F} =({\mathcal {F}}_{i})_{i\in I}} \) is a filtration, then the corresponding right-continuous filtration is defined as[2]
\( {\displaystyle \mathbb {F} ^{+}:=({\mathcal {F}}_{i}^{+})_{i\in I},} \)
with
\( {\displaystyle {\mathcal {F}}_{i}^{+}:=\bigcap _{i<z}{\mathcal {F}}_{z}.} \)
The filtration \( {\mathbb F} \) itself is called right-continuous if \( {\displaystyle \mathbb {F} ^{+}=\mathbb {F} } \).[3]
Complete filtration
Let
\( {\displaystyle {\mathcal {N}}_{P}:=\{A\subset \Omega \mid A\subset B{\text{ for some }}B{\text{ with }}P(B)=0\}} \)
be the set of all sets that are contained within a P {\displaystyle P} 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 is equivalent to \( {\displaystyle (\Omega ,{\mathcal {F}}_{i},P)} \) being a complete measure space for every \( {\displaystyle i\in I.} \)
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 \( {\displaystyle {\tilde {\mathbb {F} }}} \) of \( {\mathbb F}. \)
If a filtration is an augmented filtration, it is said to satisfy the usual hypotheses or the usual conditions.[3]
See also
Natural filtration
Filtration (mathematics)
Filter (mathematics)
References
Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 191. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 350-351. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 462. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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