filtered probability space
A filtered probability space, or stochastic basis, $(\mathrm{\Omega},\mathcal{F},{({\mathcal{F}}_{t})}_{t\in T},\mathbb{P})$ consists of a probability space^{} $(\mathrm{\Omega},\mathcal{F},\mathbb{P})$ 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 $$ 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., continuoustime), 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.

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The probability space $(\mathrm{\Omega},\mathcal{F},\mathbb{P})$ is complete^{} (http://planetmath.org/CompleteMeasure).

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The $\sigma $algebras ${\mathcal{F}}_{t}$ contain all the sets in $\mathcal{F}$ of zero probability.

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The filtration ${\mathcal{F}}_{t}$ is rightcontinuous. That is, for every nonmaximal $t\in T$, 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 rightcontinuous 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 DoobMeyer decomposition, section^{} theorems, etc.
Title  filtered probability space 

Canonical name  FilteredProbabilitySpace 
Date of creation  20130322 18:36:51 
Last modified on  20130322 18:36:51 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  5 
Author  gel (22282) 
Entry type  Definition 
Classification  msc 60G05 
Related topic  FiltrationOfSigmaAlgebras 
Defines  stochastic basis 
Defines  usual conditions 
Defines  usual hypotheses 