# Filtration (probability theory)

(Redirected from Usual hypotheses)

In the theory of stochastic processes, a subdiscipline of probability theory, filtrations 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 ${\displaystyle I}$ be an index set that is either ${\displaystyle \mathbb {N} }$, ${\displaystyle \mathbb {R} ^{+}}$, 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.

In a more general sense, filtrations can be defined for any linearly ordered index set ${\displaystyle I}$.[2]

## 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

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

## 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

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

### Complete filtration

Let

${\displaystyle {\mathcal {N}}_{P}:=\{A\subset {\mathcal {P}}(\Omega )\mid A\subset B{\text{ for a }}B{\text{ with }}P(B)=0\}}$

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 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 ${\displaystyle \mathbb {F} }$ there exists a smallest augmented filtration ${\displaystyle {\tilde {\mathbb {F} }}}$ of ${\displaystyle \mathbb {F} }$.

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