Let

• (Ω, F, P) be a probability space;
• I be an index set with a total order ≤ (often, I is N, N₀, [0, T] or [0, +∞));
• F_• = (F_i)_i∈I be a filtration of the sigma algebra F;
• (S, Σ) be a measurable space, the state space;
• X : I × Ω → S be a stochastic process.

The process X is said to be adapted to the filtration (F_i)_i∈I if the random variable X_i : Ω → S is a (F_i, Σ)-measurable function for each i ∈ I.

Consider a stochastic process X : [0, T] × 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 F_t^X is the σ-algebra generated by the pre-images X_s⁻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 : [0, 2] × Ω → R: set F_t to be the trivial σ-algebra {∅, Ω} for times 0 ≤ t < 1, and F_t = F_t^X 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 [0, 1] will fail to be F_•-adapted. The non-constant nature of such a process “uses information” from the more refined “future” σ-algebras F_t, 1 ≤ t ≤ 2.

• Progressively measurable process