In the study of stochastic processes, a stochastic process is **adapted** (also referred to as a **non-anticipating** or **non-anticipative process**) if information about the value of the process at a given time is available at that same time. 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.

##
Definition

Let

- $(\Omega ,{\mathcal {F)),\mathbb {P} )$ be a probability space;
- $I$ be an index set with a total order $\leq$ (often, $I$ is $\mathbb {N}$, $\mathbb {N} _{0))$, $[0,T]$ or $[0,+\infty )$);
- $\mathbb {F} =\left({\mathcal {F))_{i}\right)_{i\in I))$ be a filtration of the sigma algebra ${\mathcal {F))$;
- $(S,\Sigma )$ be a measurable space, the
*state space*;
- $X:I\times \Omega \to S$ 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 $X_{i}:\Omega \to S$ is a $({\mathcal {F))_{i},\Sigma )$-measurable function for each $i\in I$.^{[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 *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.