Filtration (probability theory)

Short description: Model of information available at a given point of a random process

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.

## Contents

* 1 Definition
* 2 Example
* 3 Types of filtrations
* 3.1 Right-continuous filtration
* 3.2 Complete filtration
* 3.3 Augmented filtration
* 4 See also
* 5 References

## Definition

Let (Ω,𝒜,P) be a probability space and let I be an index set with a total order ≤ (often ℕ, ℝ+, or a subset of ℝ+).

For every i∈I let ℱi be a sub-σ-algebra of 𝒜. Then

𝔽:=(ℱi)i∈I

is called a filtration, if ℱk⊆ℱℓ for all k≤ℓ. So filtrations are families of σ-algebras that are ordered non-decreasingly.[1] If 𝔽 is a filtration, then (Ω,𝒜,𝔽,P) is called a filtered probability space.

## Example

Let (Xn)n∈ℕ be a stochastic process on the probability space (Ω,𝒜,P). Let σ(Xk∣k≤n) denote the σ-algebra generated by the random variables X1,X2,…,Xn. Then

ℱn:=σ(Xk∣k≤n)

is a σ-algebra and 𝔽=(ℱn)n∈ℕ is a filtration.

𝔽 really is a filtration, since by definition all ℱn are σ-algebras and

σ(Xk∣k≤n)⊆σ(Xk∣k≤n+1).

This is known as the natural filtration of 𝒜 with respect to (Xn)n∈ℕ.

## Types of filtrations

### Right-continuous filtration

If 𝔽=(ℱi)i∈I is a filtration, then the corresponding right-continuous filtration is defined as[2]

𝔽+:=(ℱi+)i∈I,

with

ℱi+:=⋂i<zℱz.

The filtration 𝔽 itself is called right-continuous if 𝔽+=𝔽.[3]

### Complete filtration

Let (Ω,ℱ,P) be a probability space and let,

𝒩P:={A⊆Ω∣A⊆B for some B∈ℱ with P(B)=0}

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

A filtration 𝔽=(ℱi)i∈I is called a complete filtration, if every ℱi contains 𝒩P. This implies (Ω,ℱi,P) is a complete measure space for every i∈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 𝔽~ refining 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

1. ↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 191. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_341.
2. ↑ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. 77. Switzerland: Springer. p. 350-351. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
3. ↑ 3.0 3.1 Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 462. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_646.

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