# measurability of stochastic processes

For a continuous-time stochastic process adapted to a given filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) $(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}}$ on a measurable space $(\Omega,\mathcal{F})$, there are various conditions which can be placed either on its sample paths or on its measurability when considered as a function from $\mathbb{R}_{+}\times\Omega$ to $\mathbb{R}$. The following theorem lists the dependencies between these properties.

###### Theorem.

Let $(X_{t})_{t\in\mathbb{R}_{+}}$ 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. 1.

$X$ is predictable.

2. 2.

$X$ is optional.

3. 3.

$X$ is progressive.

4. 4.

$X$ is adapted and jointly measurable.

Title measurability of stochastic processes MeasurabilityOfStochasticProcesses 2013-03-22 18:37:29 2013-03-22 18:37:29 gel (22282) gel (22282) 5 gel (22282) Theorem msc 60G05 MeasurabilityOfStoppedProcesses