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adapted process

(Definition)

Let $\lbrace X_t \mid t\in T\rbrace$ be a stochastic process defined on a probability space $(\Omega,\mathcal{F},P)$ and $\lbrace \mathcal{F}_t \mid t\in T\rbrace$ a filtration (an increasing sequence of sigma subalgebras of $\mathcal{F}$ ), where is a linearly ordered subset of $\mathbb{R}$ with a minimum . Then the process $\lbrace X_t\rbrace$ is said to be adapted to the filtration $\lbrace \mathcal{F}_t\rbrace$ if for each $t\ge t_0$ , is $\mathcal{F}_t$ -measurable:

$\displaystyle X_t^{-1}(B)\in \mathcal{F}_t$ for each Borel set $\displaystyle B\in\mathbb{R}.$

A stochastic process is an adapted process if it is adapted to some filtration.

"adapted process" is owned by rspuzio. [ full author list (3) | owner history (1) ]

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Other names:

adapted

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Cross-references: subset, linearly ordered, subalgebras, sequence, increasing, filtration, probability space, stochastic process
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This is version 16 of adapted process, born on 2006-09-22, modified 2007-02-26.
Object id is 8390, canonical name is MathcalF_tMeasurableFunction.
Accessed 1968 times total.

Classification:

AMS MSC:	60A99 (Probability theory and stochastic processes :: Foundations of probability theory :: Miscellaneous)
	60G07 (Probability theory and stochastic processes :: Stochastic processes :: General theory of processes)

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