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 ${\displaystyle (\Omega ,{\mathcal {A}},P)}$. Let ${\displaystyle \sigma (X_{k}\mid k\leq n)}$ denote the σ-algebra generated by the random variables ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$. 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.

${\displaystyle \mathbb {F} }$ 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 ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$.

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.