adapted process

Let $\{X_t\mid t\in T\}$ be a stochastic process defined on a probability space $(\mathrm{\Omega },ℱ,P)$ and $\{{ℱ}_t\mid t\in T\}$ a filtration (an increasing sequence of sigma subalgebras of $ℱ$), where $T$ is a linearly ordered subset of $ℝ$ with a minimum $t_0$. Then the process $\{X_t\}$ is said to be adapted to the filtration $\{{ℱ}_t\}$ if for each $t\ge t_0$, $X_t$ is ${ℱ}_t$-measurable (http://planetmath.org/MathcalFMeasurableFunction):

$$X_t^{-1}(B)\in {ℱ}_t\text{ for each Borel set }B\in ℝ.$$

A stochastic process is an adapted process if it is adapted to some filtration.

Title	adapted process
Canonical name	AdaptedProcess
Date of creation	2013-03-22 16:16:43
Last modified on	2013-03-22 16:16:43
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	19
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 60A99
Classification	msc 60G07
Synonym	adapted