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Model of information available at a given point of a random process From Wikipedia, the free encyclopedia
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.
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.
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 .
If is a filtration, then the corresponding right-continuous filtration is defined as[2]
with
The filtration itself is called right-continuous if .[3]
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.)
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]
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