Stochastic Basis

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[edit]

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