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Stochastic basis

From Encyclopedia of Mathematics
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A complete probability space $ ( \Omega , {\mathcal F} , {\mathsf P}) $ with an increasing family $ \mathbf F = ( {\mathcal F} _ {t} ) _ {t \geq 0 } $ of sub- $ \sigma $- algebras $ {\mathcal F} _ {t} \subseteq {\mathcal F} $, which satisfies the (so-called usual) conditions:

1) it must be continuous from the right, $ {\mathcal F} _ {t} = {\mathcal F} _ {t ^ {+} } $( $ = \cap _ {s>} t {\mathcal F} _ {s} $), $ t \geq 0 $;

2) it must be complete, i.e. $ {\mathcal F} _ {t} $ contains all subsets from $ {\mathcal F} $ of $ {\mathsf P} $- measure zero.

For stochastic bases, the notations $ ( \Omega , {\mathcal F}, \mathbf F , {\mathsf P}) $ or $ ( \Omega , {\mathcal F} , ( {\mathcal F} _ {t} ) _ {t \geq 0 } , {\mathsf P}) $ are also used.

Comments

An increasing family of ( $ \sigma $-) algebras is usually called a filtration.

How to Cite This Entry:
Stochastic basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_basis&oldid=16490
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article