 Adapted process

In the study of stochastic processes, an adapted process (or nonanticipating process) is one that cannot "see into the future". An informal interpretation^{[1]} 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.
Contents
Definition
Let
 be a probability space;
 I be an index set with a total order (often, I is , , [0,T] or );
 be a filtration of the sigma algebra ;
 (S,Σ) be a measurable space, the state space;
 be a stochastic process.
The process X is said to be adapted to the filtration if the random variable is a measurable function for each .^{[2]}
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 σalgebra generated by the preimages 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 nonadapted process X : [0, 2] × Ω → R: set F_{t} to be the trivial σalgebra {∅, 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 σalgebra is to be constant, any process X that is nonconstant on [0, 1] will fail to be F_{•}adapted. The nonconstant nature of such a process "uses information" from the more refined "future" σalgebras F_{t}, 1 ≤ t ≤ 2.
See also
 Predictable process
 Progressively measurable process
References
Categories: Stochastic processes
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