Filtration (probability theory)#Augmented filtration

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

For every $i\; \backslash in\; I$ let $\backslash mathcal\; F\_i$ be a sub-σ-algebra of $\backslash mathcal\; A$. Then

:$\backslash mathbb\; F:=\; (\backslash mathcal\; F\_i)\_\{i\; \backslash in\; I\}$

is called a filtration, if $\backslash mathcal\; F\_k\; \backslash subseteq\; \backslash mathcal\; F\_\backslash ell$ for all $k\; \backslash leq\; \backslash ell$. So filtrations are families of σ-algebras that are ordered non-decreasingly. If $\backslash mathbb\; F$ is a filtration, then $(\backslash Omega,\; \backslash mathcal\; A,\; \backslash mathbb\; F,\; P)$ is called a filtered probability space.

Let $(X\_n)\_\{n\; \backslash in\; \backslash N\}$ be a stochastic process on the probability space $(\backslash Omega,\; \backslash mathcal\; A,\; P)$.

Let $\backslash sigma(X\_k\; \backslash mid\; k\; \backslash leq\; n)$ denote the σ-algebra generated by the random variables $X\_1,\; X\_2,\; \backslash dots,\; X\_n$.

Then

:$\backslash mathcal\; F\_n:=\backslash sigma(X\_k\; \backslash mid\; k\; \backslash leq\; n)$

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

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

:$\backslash sigma(X\_k\; \backslash mid\; k\; \backslash leq\; n)\; \backslash subseteq\; \backslash sigma(X\_k\; \backslash mid\; k\; \backslash leq\; n+1).$

This is known as the natural filtration of $\backslash mathcal\; A$ with respect to $(X\_n)\_\{n\; \backslash in\; \backslash N\}$.

If $\backslash mathbb\; F=\; (\backslash mathcal\; F\_i)\_\{i\; \backslash in\; I\}$ is a filtration, then the corresponding right-continuous filtration is defined as

:$\backslash mathbb\; F^+:=\; (\backslash mathcal\; F\_i^+)\_\{i\; \backslash in\; I\},$

with

:$\backslash mathcal\; F\_i^+:=\; \backslash bigcap\_\{z\; >\; i\}\; \backslash mathcal\; F\_z.$

The filtration $\backslash mathbb\; F$ itself is called right-continuous if $\backslash mathbb\; F^+\; =\; \backslash mathbb\; F$.

Let $(\backslash Omega,\; \backslash mathcal\; F,\; P)$ be a probability space, and let

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

be the set of all sets that are contained within a $P$-null set.

A filtration $\backslash mathbb\; F=\; (\backslash mathcal\; F\_i)\_\{i\; \backslash in\; I\}$ is called a complete filtration, if every $\backslash mathcal\; F\_i$ contains $\backslash mathcal\; N\_P$. This implies $(\backslash Omega,\; \backslash mathcal\; F\_i,\; P)$ is a complete measure space for every $i\; \backslash in\; I.$ (The converse is not necessarily true.)

A filtration is called an augmented filtration if it is complete and right continuous. For every filtration $\backslash mathbb\; F$ there exists a smallest augmented filtration $\backslash tilde\; \{\backslash mathbb\; F\}$ refining $\backslash mathbb\; F$.

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

• Natural filtration
• Filtration (mathematics)
• Filter (mathematics)

{{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|series=Probability Theory and Stochastic Modelling |volume=77 |location= Switzerland |publisher=Springer |doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3|page=350-351}}

{{cite book |last1=Klenke |first1=Achim |year=2008 |title=Probability Theory |url=https://archive.org/details/probabilitytheor00klen_341 |url-access=limited |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6|page=[https://archive.org/details/probabilitytheor00klen_341/page/n196 191] }}

{{cite book |last1=Klenke |first1=Achim |year=2008 |title=Probability Theory |url=https://archive.org/details/probabilitytheor00klen_646 |url-access=limited |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6|page=[https://archive.org/details/probabilitytheor00klen_646/page/n461 462] }}

Category:Probability theory