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 ${\displaystyle (\mathrm{\Omega },\mathcal{A},P)}$ be a probability space and let ${\displaystyle I}$ be an index set with a total order ${\displaystyle \le }$ (often ${\displaystyle \text{N}}$, ${\displaystyle {\text{R}}^{+}}$, 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 ${\displaystyle k\le \ell }$. So filtrations are families of σ-algebras that are ordered non-decreasingly.^[1] If ${\displaystyle \mathbb{F}}$ is a filtration, then ${\displaystyle (\mathrm{\Omega },\mathcal{A},\mathbb{F},P)}$ is called a filtered probability space.

Example

Let ${\displaystyle (X_n{)}_{n\in \text{N}}}$ be a stochastic process on the probability space ${\displaystyle (\mathrm{\Omega },\mathcal{A},P)}$. Let ${\displaystyle \sigma (X_k\mid k\le n)}$ denote the σ-algebra generated by the random variables ${\displaystyle X_1,X_2,\dots ,X_n}$. Then

${\displaystyle {\mathcal{F}}_n:=\sigma (X_k\mid k\le n)}$

is a σ-algebra and ${\displaystyle \mathbb{F}=({\mathcal{F}}_n{)}_{n\in \text{N}}}$ is a filtration.

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

${\displaystyle \sigma (X_k\mid k\le n)\subseteq \sigma (X_k\mid k\le n+1).}$

This is known as the natural filtration of ${\displaystyle \mathcal{A}}$ with respect to ${\displaystyle (X_n{)}_{n\in \text{N}}}$.

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

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

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 ${\displaystyle (\mathrm{\Omega },{\mathcal{F}}_i,P)}$ 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

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