Filtration (probability theory)

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

For every ${\displaystyle i\in I}$  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 ${\displaystyle {\mathcal {F}}_{k}\subseteq {\mathcal {F}}_{\ell }}$  for all ${\displaystyle k\leq \ell }$ . So filtrations are families of σ-algebras that are ordered non-decreasingly.^[1] If ${\displaystyle \mathbb {F} }$  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 ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ .

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

${\displaystyle \sigma (X_{k}\mid k\leq n)\subseteq \sigma (X_{k}\mid k\leq n+1).}$ 

This is known as the natural filtration of ${\displaystyle {\mathcal {A}}}$  with respect to ${\displaystyle X}$ .

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

Complete filtration

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

${\displaystyle {\mathcal {N}}_{P}:=\{A\subseteq \Omega \mid A\subseteq B{\text{ for some }}B\in {\mathcal {F}}{\text{ with }}P(B)=0\}}$ 

be the set of all sets that are contained within a ${\displaystyle P}$ -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 implies ${\displaystyle (\Omega ,{\mathcal {F}}_{i},P)}$  is a complete measure space for every ${\displaystyle i\in I.}$  (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 ${\displaystyle \mathbb {F} }$  there exists a smallest augmented filtration ${\displaystyle {\tilde {\mathbb {F} }}}$  refining ${\displaystyle \mathbb {F} }$ .

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

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