Filtration (probability theory)

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. 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 This is known as the natural filtration of with respect to . If is a filtration, then the corresponding right-continuous filtration is defined as with The filtration itself is called right-continuous if . 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.