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.
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. If is a filtration, then is called a filtered probability space.
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 filtrations
If is a filtration, then the corresponding right-continuous filtration is defined as
The filtration itself is called right-continuous iff .
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 .
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 a augmented filtration, it is said to satisfy the usual hypotheses or the usual conditions
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- 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.
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