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- Natural_filtration abstract "In the theory of stochastic processes in mathematics and statistics, the 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; let (I, ≤) be a totally ordered index set; let (S, Σ) be a measurable space; 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 = (FiX)i∈I given byi.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 [0, T] or [0, +∞); the state space S is often the real line R or Euclidean space Rn.Any stochastic process X is an adapted process with respect to its natural filtration.".
- Natural_filtration wikiPageID "5864648".
- Natural_filtration wikiPageRevisionID "586740810".
- Natural_filtration hasPhotoCollection Natural_filtration.
- Natural_filtration subject Category:Stochastic_processes.
- Natural_filtration type Abstraction100002137.
- Natural_filtration type Cognition100023271.
- Natural_filtration type Concept105835747.
- Natural_filtration type Content105809192.
- Natural_filtration type Hypothesis105888929.
- Natural_filtration type Idea105833840.
- Natural_filtration type Model105890249.
- Natural_filtration type PsychologicalFeature100023100.
- Natural_filtration type StochasticProcess113561896.
- Natural_filtration type StochasticProcesses.
- Natural_filtration comment "In the theory of stochastic processes in mathematics and statistics, the natural filtration associated to a stochastic process is a filtration associated to the process which records its "past behaviour" at each time.".
- Natural_filtration label "Natural filtration".
- Natural_filtration sameAs m.0f9rdv.
- Natural_filtration sameAs Q6980678.
- Natural_filtration sameAs Q6980678.
- Natural_filtration sameAs Natural_filtration.
- Natural_filtration wasDerivedFrom Natural_filtration?oldid=586740810.
- Natural_filtration isPrimaryTopicOf Natural_filtration.