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 $(\mathrm{\Omega },{𝒜},P)$ be a probability space and let $I$ be an index set with a total order $\le$ (often $\mathbb{\mathbb{N}}$, ${\mathbb{ℝ}}^{+}$, or a subset of ${\mathbb{ℝ}}^{+}$). For every $i\in I$ let ${{ℱ}}_i$ be a sub-σ-algebra of ${𝒜}$. Then

$\mathbb{𝔽}:=({{ℱ}}_i{)}_{i\in I}$

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 $\mathbb{𝔽}$ is a filtration, then $(\mathrm{\Omega },{𝒜},\mathbb{𝔽},P)$ is called a filtered probability space.

Example

Let $(X_n{)}_{n\in \mathbb{\mathbb{N}}}$ 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 $\mathbb{𝔽}=({{ℱ}}_n{)}_{n\in \mathbb{\mathbb{N}}}$ is a filtration. $\mathbb{𝔽}$ 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 $(X_n{)}_{n\in \mathbb{\mathbb{N}}}$.

Types of filtrations

Right-continuous filtration

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

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

with

${{ℱ}}_i^{+}:=\bigcap _{i<z}{{ℱ}}_z.$

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

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 $\mathbb{𝔽}=({{ℱ}}_i{)}_{i\in I}$ 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 $\mathbb{𝔽}$ there exists a smallest augmented filtration $\widetilde{\mathbb{𝔽}}$ refining $\mathbb{𝔽}$. 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)