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.
$X$ is predictable.
2.
$X$ is optional.
3.
$X$ is progressive.
4.
$X$ is adapted and jointly measurable.

单词	MeasurabilityOfStochasticProcesses
释义	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. $X$ is predictable. 2. $X$ is optional. 3. $X$ is progressive. 4. $X$ is adapted and jointly measurable.
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