Adapted process

Definition

Let

• (\Omega, \mathcal{F}, \mathbb{P}) be a probability space;
• I be an index set with a total order \leq (often, I is \mathbb{N}, \mathbb{N}_0, [0, T] or 0, +\infty));
• \left(\mathcal{F}i\right){i \in I} be a [filtration of the sigma algebra \mathcal{F};
• (S,\Sigma) be a measurable space, the state space;
• X_i: I \times \Omega \to S be a stochastic process.

The stochastic process (X_i)_{i\in I} is said to be adapted to the filtration \left(\mathcal{F}i\right){i \in I} if the random variable X_i: \Omega \to S is a (\mathcal{F}_i, \Sigma)-measurable function for each i \in I.

Examples

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 FtX 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 t = 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 [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.

References

References

1. Wiliams, David. (1979). "Diffusions, Markov Processes and Martingales: Foundations". Wiley.
2. Øksendal, Bernt. (2003). "Stochastic Differential Equations". Springer.