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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]

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.