In the study of
stochastic processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
, 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'', ''X
n'' is known at time ''n''. The concept of an adapted process is essential, for instance, in the definition of the
Itō integral
Itō may refer to:
*Itō (surname), a Japanese surname
*Itō, Shizuoka, Shizuoka Prefecture, Japan
*Ito District, Wakayama Prefecture, Japan
See also
* Itô's lemma, used in stochastic calculus
*Itoh–Tsujii inversion algorithm, in field theory ...
, which only makes sense if the
integrand
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
is an adapted process.
Definition
Let
*
be a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
;
*
be an index set with a total order
(often,
is
,
,