measurability of stochastic processes

For a continuous-time stochastic process adapted to a given filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) ${({ℱ}_t)}_{t\in {ℝ}_{+}}$ on a measurable space $(\mathrm{\Omega },ℱ)$, there are various conditions which can be placed either on its sample paths or on its measurability when considered as a function from ${ℝ}_{+}\times \mathrm{\Omega }$ to $ℝ$. 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 right-continuous, it is predictable if it is adapted and left-continuous. 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	2013-03-22 18:37:29
Last modified on	2013-03-22 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