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; let (I, ≤) be a totally orderedindex set; let (S, Σ) be a measurable space; let X : I × Ω → S be a stochastic process. Then the natural filtration ofFwith respect toX is defined to be the filtration F•X = (FiX)i∈I given by
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.
natural, filtration, theory, stochastic, processes, mathematics, statistics, generated, filtration, natural, filtration, associated, stochastic, process, filtration, associated, process, which, records, past, behaviour, each, time, sense, simplest, filtration,. 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 W F P be a probability space let I be a totally ordered index set let S S be a measurable space let X I W 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 by F i X s X j 1 A j I j i A S displaystyle F i X sigma left left X j 1 A right j in I j leq i A in Sigma right i e the smallest s algebra on W that contains all pre images of S 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 References EditDelia Coculescu Ashkan Nikeghbali 2010 Filtrations Encyclopedia of Quantitative FinanceSee also EditFiltration mathematics Retrieved from https en wikipedia org w index php title Natural filtration amp oldid 868556916, wikipedia, wiki, book, books, library,