# Adapted process

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

- [math]\displaystyle{ (\Omega, \mathcal{F}, \mathbb{P}) }[/math] be a probability space;
- [math]\displaystyle{ I }[/math] be an index set with a total order [math]\displaystyle{ \leq }[/math] (often, [math]\displaystyle{ I }[/math] is [math]\displaystyle{ \mathbb{N} }[/math], [math]\displaystyle{ \mathbb{N}_0 }[/math], [math]\displaystyle{ [0, T] }[/math] or [math]\displaystyle{ [0, +\infty) }[/math]);
- [math]\displaystyle{ \mathbb F = \left(\mathcal{F}_i\right)_{i \in I} }[/math] be a filtration of the sigma algebra [math]\displaystyle{ \mathcal{F} }[/math];
- [math]\displaystyle{ (S,\Sigma) }[/math] be a measurable space, the
*state space*; - [math]\displaystyle{ X: I \times \Omega \to S }[/math] be a stochastic process.

The process [math]\displaystyle{ X }[/math] is said to be **adapted to the filtration** [math]\displaystyle{ \left(\mathcal{F}_i\right)_{i \in I} }[/math] if the random variable [math]\displaystyle{ X_i: \Omega \to S }[/math] is a [math]\displaystyle{ (\mathcal{F}_i, \Sigma) }[/math]-measurable function for each [math]\displaystyle{ i \in I }[/math].^{[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.

## See also

## References

- ↑ Wiliams, David (1979).
*Diffusions, Markov Processes and Martingales: Foundations*.**1**. Wiley. ISBN 0-471-99705-6. - ↑ Øksendal, Bernt (2003).
*Stochastic Differential Equations*. Springer. p. 25. ISBN 978-3-540-04758-2.

Original source: https://en.wikipedia.org/wiki/Adapted process.
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