Filtration (probability theory)

Template:Short description 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 $(\mathrm{\Omega },𝒜,P)$ be a probability space and let $I$ be an index set with a total order $\le$ (often $\mathbb{N}$, ${ℝ}^{+}$, or a subset of ${ℝ}^{+}$).

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

$𝔽:=({ℱ}_i{)}_{i\in I}$

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

Example

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

${ℱ}_n:=\sigma (X_k\mid k\le n)$

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

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

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

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

Types of filtrations

Right-continuous filtration

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

${𝔽}^{+}:=({ℱ}_i^{+}{)}_{i\in I},$

with

${ℱ}_i^{+}:=\bigcap _{z>i}{ℱ}_z.$

The filtration $𝔽$ itself is called right-continuous if ${𝔽}^{+}=𝔽$.

Complete filtration

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

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

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

A filtration $𝔽=({ℱ}_i{)}_{i\in I}$ is called a complete filtration, if every ${ℱ}_i$ contains ${𝒩}_P$. This implies $(\mathrm{\Omega },{ℱ}_i,P)$ is a complete measure space for every $i\in I.$ (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 $𝔽$ there exists a smallest augmented filtration $\widetilde{𝔽}$ refining $𝔽$.

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

See also

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