Cameron–Martin theorem

In the study of stochastic processes, an adapted process (or non-anticipating process) is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, X_n is known at time n. The concept of an adapted process is essential, for instance, in the definition of the Itō integral, which only makes sense if the integrand is an adapted process.

Definition

Let

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

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

See also

• Predictable process
• Progressively measurable process

References

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