Adapted process
In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative 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, Xn 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
be a probability space;
be an index set with a total order
(often,
is
,
,
or
);
be a filtration of the sigma algebra
;
be a measurable space, the state space;
be a stochastic process.
The process
is said to be adapted to the filtration
if the random variable
is a
-measurable function for each
.
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 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.