filtered probability space

A filtered probability space, or stochastic basis, $(\\Omega,\\mathcal{F},(\\mathcal{F}_{t})_{t\\in T},\\mathbb{P})$ consists of a probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$ and a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) $(\\mathcal{F}_{t})_{t\\in T}$ contained in $\\mathcal{F}$ . Here, $T$ is the time index set, and is an ordered set — usually a subset of the real numbers — such that $\\mathcal{F}_{s}\\subseteq\\mathcal{F}_{t}$ for all $s<t$ in $T$ .

Filtered probability spaces form the setting for defining and studying stochastic processes. A process $X_{t}$ with time index $t$ ranging over $T$ is said to be adapted if $X_{t}$ is an $\\mathcal{F}_{t}$ -measurable random variable for every $t$ .

When the index set $T$ is an interval (http://planetmath.org/Interval) of the real numbers (i.e., continuous-time), it is often convenient to impose further conditions. In this case, the filtered probability space is said to satisfy the usual conditions or usual hypotheses if the following conditions are met.

•
The probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$ is complete (http://planetmath.org/CompleteMeasure).
•
The $\\sigma$ -algebras $\\mathcal{F}_{t}$ contain all the sets in $\\mathcal{F}$ of zero probability.
•
The filtration $\\mathcal{F}_{t}$ is right-continuous. That is, for every non-maximal $t\\in T$ , the $\\sigma$ -algebra $\\mathcal{F}_{t+}\\equiv\\bigcap_{s>t}\\mathcal{F}_{s}$ is equal to $\\mathcal{F}_{t}$ .

Given any filtered probability space, it can always be enlarged by passing to the completion of the probability space, adding zero probability sets to $\\mathcal{F}_{t}$ , and by replacing $\\mathcal{F}_{t}$ by $\\mathcal{F}_{t+}$ . This will then satisfy the usual conditions.In fact, for many types of processes defined on a complete probability space, their natural filtration will already be right-continuous and the usual conditions met.However, the process of completing the probability space depends on the specific probability measure $\\mathbb{P}$ and in many situations, such as the study of Markov processes, it is necessary to study many different measures on the same space. A much weaker condition which can be used is that the $\\sigma$ -algebras $\\mathcal{F}_{t}$ are universally complete, which is still strong enough to apply much of the ‘heavy machinery’ of stochastic processes, such as the Doob-Meyer decomposition, section theorems, etc.

单词	FilteredProbabilitySpace
释义	filtered probability space A filtered probability space, or stochastic basis, $(\\Omega,\\mathcal{F},(\\mathcal{F}_{t})_{t\\in T},\\mathbb{P})$ consists of a probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$ and a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) $(\\mathcal{F}_{t})_{t\\in T}$ contained in $\\mathcal{F}$ . Here, $T$ is the time index set, and is an ordered set — usually a subset of the real numbers — such that $\\mathcal{F}_{s}\\subseteq\\mathcal{F}_{t}$ for all $s<t$ in $T$ . Filtered probability spaces form the setting for defining and studying stochastic processes. A process $X_{t}$ with time index $t$ ranging over $T$ is said to be adapted if $X_{t}$ is an $\\mathcal{F}_{t}$ -measurable random variable for every $t$ . When the index set $T$ is an interval (http://planetmath.org/Interval) of the real numbers (i.e., continuous-time), it is often convenient to impose further conditions. In this case, the filtered probability space is said to satisfy the usual conditions or usual hypotheses if the following conditions are met. • The probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$ is complete (http://planetmath.org/CompleteMeasure). • The $\\sigma$ -algebras $\\mathcal{F}_{t}$ contain all the sets in $\\mathcal{F}$ of zero probability. • The filtration $\\mathcal{F}_{t}$ is right-continuous. That is, for every non-maximal $t\\in T$ , the $\\sigma$ -algebra $\\mathcal{F}_{t+}\\equiv\\bigcap_{s>t}\\mathcal{F}_{s}$ is equal to $\\mathcal{F}_{t}$ . Given any filtered probability space, it can always be enlarged by passing to the completion of the probability space, adding zero probability sets to $\\mathcal{F}_{t}$ , and by replacing $\\mathcal{F}_{t}$ by $\\mathcal{F}_{t+}$ . This will then satisfy the usual conditions.In fact, for many types of processes defined on a complete probability space, their natural filtration will already be right-continuous and the usual conditions met.However, the process of completing the probability space depends on the specific probability measure $\\mathbb{P}$ and in many situations, such as the study of Markov processes, it is necessary to study many different measures on the same space. A much weaker condition which can be used is that the $\\sigma$ -algebras $\\mathcal{F}_{t}$ are universally complete, which is still strong enough to apply much of the ‘heavy machinery’ of stochastic processes, such as the Doob-Meyer decomposition, section theorems, etc.
随便看	uniform module UniformNeighborhood uniform neighborhood UniformProximityIsAProximity uniform proximity is a proximity UniformSpace uniform space UniformStructureOfAMetricSpace uniform structure of a metric space UniformStructureOfATopologicalGroup uniform structure of a topological group Unimodular unimodular UnimodularGroup unimodular group UnimodularLattice unimodular lattice UnimodularMatrix unimodular matrix Union union UnionOfNondisjointConnectedSetsIsConnected union of non-disjoint connected sets is connected UniqueFactorizationAndIdealsInRingOfIntegers unique factorization and ideals in ring of integers