measurability of stochastic processes
For a continuoustime stochastic process adapted to a given filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) ${({\mathcal{F}}_{t})}_{t\in {\mathbb{R}}_{+}}$ on a measurable space^{} $(\mathrm{\Omega},\mathcal{F})$, there are various conditions which can be placed either on its sample paths or on its measurability when considered as a function from ${\mathbb{R}}_{+}\times \mathrm{\Omega}$ to $\mathbb{R}$. The following theorem lists the dependencies between these properties.
Theorem.
Let ${\mathrm{(}{X}_{t}\mathrm{)}}_{t\mathrm{\in}{\mathrm{R}}_{\mathrm{+}}}$ be a real valued stochastic process^{}. Then, $X$ is optional if it is adapted and rightcontinuous, it is predictable if it is adapted and leftcontinuous. Furthermore, each of the following properties implies the next.

1.
$X$ is predictable.

2.
$X$ is optional.

3.
$X$ is progressive.

4.
$X$ is adapted and jointly measurable.
Title  measurability of stochastic processes 

Canonical name  MeasurabilityOfStochasticProcesses 
Date of creation  20130322 18:37:29 
Last modified on  20130322 18:37:29 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  5 
Author  gel (22282) 
Entry type  Theorem 
Classification  msc 60G05 
Related topic  MeasurabilityOfStoppedProcesses 