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Definition

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

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

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

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

Example

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

F n := σ ( X k ∣ k ≤ n ) {\displaystyle {\mathcal {F}}_{n}:=\sigma (X_{k}\mid k\leq n)}

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

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

σ ( X k ∣ k ≤ n ) ⊆ σ ( X k ∣ k ≤ 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 A {\displaystyle {\mathcal {A}}} with respect to ( X n ) n ∈ N {\displaystyle (X_{n})_{n\in \mathbb {N} }} .

Types of filtrations

Right-continuous filtration

If F = ( F i ) i ∈ I {\displaystyle \mathbb {F} =({\mathcal {F}}_{i})_{i\in I}} is a filtration, then the corresponding right-continuous filtration is defined as²

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

with

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

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

Complete filtration

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

N P := { A ⊆ Ω ∣ A ⊆ B  for some  B ∈ F  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 P {\displaystyle P} -null set.

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

If a filtration is an augmented filtration, it is said to satisfy the usual hypotheses or the usual conditions.⁴

See also

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

References

1. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 191. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. 978-1-84800-047-6 ↩

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

3. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 462. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. 978-1-84800-047-6 ↩

4. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 462. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. 978-1-84800-047-6 ↩