Natural 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. 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 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 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 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.