Examples
Consider a stochastic process X : × Ω → R, and equip the real line R with its usual Borel sigma algebra generated by the open sets.
- If we take the natural filtration F•X, where FtX is the σ-algebra generated by the pre-images Xs−1(B) for Borel subsets B of R and times 0 ≤ s ≤ t, then X is automatically F•X-adapted. Intuitively, the natural filtration F•X contains "total information" about the behaviour of X up to time t.
- This offers a simple example of a non-adapted process X : × Ω → R: set Ft to be the trivial σ-algebra {∅, Ω} for times 0 ≤ t < 1, and Ft = FtX for times 1 ≤ t ≤ 2. Since the only way that a function can be measurable with respect to the trivial σ-algebra is to be constant, any process X that is non-constant on will fail to be F•-adapted. The non-constant nature of such a process "uses information" from the more refined "future" σ-algebras Ft, 1 ≤ t ≤ 2.
Read more about this topic: Adapted Process
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