In the theory of stochastic processes in mathematics and statistics, the generated filtration or natural filtration associated to a stochastic process is a filtration associated to the process which records its "past behaviour" at each time. It is in a sense the simplest filtration available for studying the given process: all information concerning the process, and only that information, is available in the natural filtration. More formally, let (Ω, ''F'', P) be a

probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...

; let (''I'', ≤) be a

totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...

index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...

; let (''S'', Σ) be a

measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...

; let ''X'' : ''I'' × Ω → ''S'' be a stochastic process. Then the natural filtration of ''F'' with respect to ''X'' is defined to be the filtration ''F''_•^''X'' = (''F''_''i''^''X'')_{''i''∈''I''} given by :

F_^ = \sigma \left\,

i.e., the smallest ''σ''-algebra on Ω that contains all pre-images of Σ-measurable subsets of ''S'' for "times" ''j'' up to ''i''. In many examples, the index set ''I'' is the natural numbers N (possibly including 0) or an interval , ''T''or ,_+∞);_the_state_space_''S''_is_often_the_real_line_R_or_Euclidean_space.html" ;"title="real_line.html" ;"title=", +∞); the state space ''S'' is often the real line">, +∞); the state space ''S'' is often the real line R or Euclidean space">real_line.html" ;"title=", +∞); the state space ''S'' is often the real line">, +∞); the state space ''S'' is often the real line R or Euclidean space R^''n''. Any stochastic process ''X'' is an adapted process with respect to its natural filtration.

References

See also