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 ( 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
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. 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.
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
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
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} }} .
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 as
with
The filtration F {\displaystyle \mathbb {F} } itself is called right-continuous if F + = F {\displaystyle \mathbb {F} ^{+}=\mathbb {F} } .
Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space, and let
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.)
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.
Natural filtration
Filtration (mathematics)
Filter (mathematics)
