Filtration (probability theory)

Short description: Model of information available at a given point of a random process

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

For every [math]\displaystyle{ i \in I }[/math] let [math]\displaystyle{ \mathcal F_i }[/math] be a sub-σ-algebra of [math]\displaystyle{ \mathcal A }[/math]. Then

[math]\displaystyle{ \mathbb F:= (\mathcal F_i)_{i \in I} }[/math]

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

Example

Let [math]\displaystyle{ (X_n)_{n \in \N} }[/math] be a stochastic process on the probability space [math]\displaystyle{ (\Omega, \mathcal A, P) }[/math]. Let [math]\displaystyle{ \sigma(X_k \mid k \leq n) }[/math] denote the σ-algebra generated by the random variables [math]\displaystyle{ X_1, X_2, \dots, X_n }[/math]. Then

[math]\displaystyle{ \mathcal F_n:=\sigma(X_k \mid k \leq n) }[/math]

is a σ-algebra and [math]\displaystyle{ \mathbb F= (\mathcal F_n)_{n \in \N} }[/math] is a filtration.

[math]\displaystyle{ \mathbb F }[/math] really is a filtration, since by definition all [math]\displaystyle{ \mathcal F_n }[/math] are σ-algebras and

[math]\displaystyle{ \sigma(X_k \mid k \leq n) \subseteq \sigma(X_k \mid k \leq n+1). }[/math]

This is known as the natural filtration of [math]\displaystyle{ \mathcal A }[/math] with respect to [math]\displaystyle{ (X_n)_{n \in \N} }[/math].

Types of filtrations

Right-continuous filtration

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

[math]\displaystyle{ \mathbb F^+:= (\mathcal F_i^+)_{i \in I}, }[/math]

with

[math]\displaystyle{ \mathcal F_i^+:= \bigcap_{i \lt z} \mathcal F_z. }[/math]

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

Complete filtration

Let [math]\displaystyle{ (\Omega, \mathcal F, P) }[/math] be a probability space and let,

[math]\displaystyle{ \mathcal N_P:= \{A \subseteq \Omega \mid A \subseteq B \text{ for some } B \in \mathcal F \text{ with } P(B)=0 \} }[/math]

be the set of all sets that are contained within a [math]\displaystyle{ P }[/math]-null set.

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

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

References

↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 191. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_341.
↑ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. 77. Switzerland: Springer. p. 350-351. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
↑ ^3.0 ^3.1 Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 462. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_646.

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[Klenke191-1] Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 191. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_341.

[Kallenberg350-2] Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. 77. Switzerland: Springer. p. 350-351. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.

[Klenke462-3] 3.0 ^3.1 Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 462. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_646.

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