# Adapted process

Jump to navigation
Jump to search

In the study of stochastic processes, an **adapted process** (or **non-anticipating process**) is one that cannot "see into the future". 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.

## Contents

## Definition

Let

- be a probability space;
- be an index set with a total order (often, is , , or );
- be a filtration of the sigma algebra ;
- be a measurable space, the
*state space*; - be a stochastic process.

The process is said to be **adapted to the filtration** if the random variable is a -measurable function for each .^{[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.