Filtration (probability theory)

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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.

Definition[edit]

Let be a probability space and let be an index set that is either , , 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.

In a more general sense, filtrations can be defined for any linearly ordered index set .[2]

Example[edit]

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[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 iff .[3]

Complete filtration[edit]

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 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 of .

If a filtration is a 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. ^ a b 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.