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Natural filtration

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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• = (Fi)i∈I given by

F i X = σ { X j − 1 ( A ) | j ∈ I , j ≤ i , A ∈ Σ } , {\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 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 R.

Any stochastic process X is an adapted process with respect to its natural filtration.

References

  • Delia Coculescu; Ashkan Nikeghbali (2010), "Filtrations", Encyclopedia of Quantitative Finance

See also

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