filtered probability space
A filtered probability space, or stochastic basis, consists of a probability space and a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) contained in . Here, is the time index set, and is an ordered set — usually a subset of the real numbers — such that for all in .
Filtered probability spaces form the setting for defining and studying stochastic processes. A process with time index ranging over is said to be adapted if is an -measurable random variable for every .
When the index set 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 is complete (http://planetmath.org/CompleteMeasure).
The -algebras contain all the sets in of zero probability.
The filtration is right-continuous. That is, for every non-maximal , the -algebra is equal to .
Given any filtered probability space, it can always be enlarged by passing to the completion of the probability space, adding zero probability sets to , and by replacing by . 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 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 -algebras 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.
|Title||filtered probability space|
|Date of creation||2013-03-22 18:36:51|
|Last modified on||2013-03-22 18:36:51|
|Last modified by||gel (22282)|