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measurability of stochastic processes
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(Theorem)
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For a continuous-time stochastic process adapted to a given filtration $(\mathcal{F}_t)_{t\in\mathbb{R}_+}$ 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.
- $X$ is predictable.
- $X$ is optional.
- $X$ is progressive.
- $X$ is adapted and jointly measurable.
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"measurability of stochastic processes" is owned by gel.
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Cross-references: jointly measurable, progressive, implies, predictable, optional, real, function, sample paths, measurable space, adapted, stochastic process
There are 2 references to this entry.
This is version 2 of measurability of stochastic processes, born on 2008-12-20, modified 2008-12-20.
Object id is 11361, canonical name is MeasurabilityOfStochasticProcesses.
Accessed 421 times total.
Classification:
| AMS MSC: | 60G05 (Probability theory and stochastic processes :: Stochastic processes :: Foundations of stochastic processes) |
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Pending Errata and Addenda
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