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

• be a probability space
  Probability space
  In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
  
  ;
• be an index set with a total order (often, is , , or );
• be a filtration
  Filtration (abstract algebra)
  In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj...
  
   of the sigma algebra ;
• be a measurable space, the state space;
• be a stochastic process
  Stochastic process
  In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
  
  .

The process is said to be adapted to the filtration if the random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

  is a -measurable function

Measurable function

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

 for each .

Examples

Consider a stochastic process X : [0, T] × Ω → R, and equip the real line

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

 R with its usual Borel sigma algebra generated by the open sets.

• If we take the natural filtration
  Natural filtration
  In the theory of stochastic processes in mathematics and statistics, the natural filtration associated to a stochastic process is a filtration associated to the process which records its "past behaviour" at each time...
  
  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.