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, 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

• $(\Omega, \mathcal{F}, \mathbb{P})$ 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 $[0, +\infty)$);
• $\mathcal{F}_{\cdot} = \left(\mathcal{F}_i\right)_{i \in I}$ be a filtration of the sigma algebra $\mathcal{F}$;
• (S,Σ) 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 $i \in I$.

## 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 FX, where FtX is the σ-algebra generated by the pre-images Xs−1(B) for Borel subsets B of R and times 0 ≤ st, then X is automatically FX-adapted. Intuitively, the natural filtration FX 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 Ft to be the trivial σ-algebra {∅, Ω} for times 0 ≤ t < 1, and Ft = FtX 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 Ft, 1 ≤ t ≤ 2.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Adapted Physical Education — is a sub discipline of physical education. It is an individualized program created for students who require a specially designed program for more than 30 days. The program involves physical fitness, motor fitness, fundamental motor skills and… …   Wikipedia

• Process (engineering) — Process engineering refers to engineering which is collaborative and concerned with completing a project as a whole.emiconductor devicesIn the electronics industry, especially for those building ICs, some technologists can be referred to as… …   Wikipedia

• Process (systems engineering) — See also Process (disambiguation). CPRET Systems engineering CPRET A Process Definition according to AFIS (Association Française d Ingénierie Système) dedicated to SE and open to all domains. IntroductionThe System Engineering normative documents …   Wikipedia

• process in rem — A process, as that of a court of admiralty, adapted to the establishment and enforcement of a right in a thing, as distinguished from the enforcement of a personal liability. State v Voorhies, 39 La Ann 499, 2 So 37 …   Ballentine's law dictionary

• Progressively measurable process — In mathematics, progressive measurability is a property of stochastic processes. A progressively measurable process cannot see into the future , but being progressively measurable is a strictly stronger property than the notion of being an… …   Wikipedia

• Consistent pricing process — A consistent pricing process (CPP) is any representation of (frictionless) prices of assets in a market. It is a stochastic process in a filtered probability space such that at time t the ith component can be thought of as a price for the ith… …   Wikipedia

• Meta-Process Modeling — is a type of metamodeling used in software engineering and systems engineering for the analysis and construction of models applicable and useful some predefined problems. Meta process support the effort of creating flexible process models. The… …   Wikipedia

• Meta-process modeling — Abstraction level for processes. Meta process modeling is a type of metamodeling used in software engineering and systems engineering for the analysis and construction of models applicable and useful to some predefined problems. Meta process… …   Wikipedia

• Wiener process — In mathematics, the Wiener process is a continuous time stochastic process named in honor of Norbert Wiener. It is often called Brownian motion, after Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with… …   Wikipedia

• Collodion process — An old deteriorated wet plate featuring Theodore Roosevelt The collodion process is an early photographic process. It was introduced in the 1850s and by the end of that decade it had almost entirely replaced the first practical photographic… …   Wikipedia