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
editLet 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
editLet 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
editRight-continuous filtration
editIf 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
editLet 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
editA 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
editReferences
edit- ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 191. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
- ^ 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.
- ^ 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.