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 ( 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. 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
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