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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'' = (''F''''i''''X'')''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''. In many examples, the index set ''I'' is the natural numbers N (possibly including 0) or an interval (''T'' ) or [0, +∞); 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== * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Natural filtration」の詳細全文を読む スポンサード リンク
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