Filtration (probability theory)

Summary

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

edit

Let   be a probability space and let   be an index set with a total order   (often  ,  , or a subset of  ).

For every   let   be a sub-σ-algebra of  . Then

 

is called a filtration, if   for all  . So filtrations are families of σ-algebras that are ordered non-decreasingly.[1] If   is a filtration, then   is called a filtered probability space.

Example

edit

Let   be a stochastic process on the probability space  . Let   denote the σ-algebra generated by the random variables  . Then

 

is a σ-algebra and   is a filtration.

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

 

This is known as the natural filtration of   with respect to  .

Types of filtrations

edit

Right-continuous filtration

edit

If   is a filtration, then the corresponding right-continuous filtration is defined as[2]

 

with

 

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

Complete filtration

edit

Let   be a probability space and let,

 

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

A filtration   is called a complete filtration, if every   contains  . This implies   is a complete measure space for every   (The converse is not necessarily true.)

Augmented filtration

edit

A filtration is called an augmented filtration if it is complete and right continuous. For every filtration   there exists a smallest augmented filtration   refining  .

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

See also

edit

References

edit
  1. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 191. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  2. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 350-351. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
  3. ^ a b Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 462. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.