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Filtration (probability theory)

In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are used to model the information that is available at a given point and therefore play an important role in the formalization of random processes.

DefinitionEdit

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.

ExampleEdit

Let   be a stochastic process on the probability space  . Then

 

is a σ-algebra and   is a filtration. Here   denotes the σ-algebra generated by the random variables  .

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

 

Types of filtrationsEdit

Right-continuous filtrationEdit

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

 

with

 

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

Complete filtrationEdit

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 is equivalent to   being a complete measure space for every  

Augmented filtrationEdit

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

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

See alsoEdit

ReferencesEdit

  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. 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.