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Difference between revisions of "Stochastic basis"

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A complete probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s0900201.png" /> with an increasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s0900202.png" /> of sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s0900203.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s0900204.png" />, which satisfies the (so-called usual) conditions:
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1) it must be continuous from the right, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s0900205.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s0900206.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s0900207.png" />;
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2) it must be complete, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s0900208.png" /> contains all subsets from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s0900209.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s09002010.png" />-measure zero.
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A complete probability space  $  ( \Omega , {\mathcal F} , {\mathsf P}) $
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with an increasing family  $  \mathbf F = ( {\mathcal F} _ {t} ) _ {t \geq  0 }  $
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of sub- $  \sigma $-
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algebras  $  {\mathcal F} _ {t} \subseteq {\mathcal F} $,
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which satisfies the (so-called usual) conditions:
  
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For stochastic bases, the notations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s09002011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s09002012.png" /> are also used.
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1) it must be continuous from the right, $  {\mathcal F} _ {t} = {\mathcal F} _ {t  ^ {+}  } $(
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$  = \cap _ {s>} t {\mathcal F} _ {s} $),
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$  t \geq  0 $;
  
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2) it must be complete, i.e.  $  {\mathcal F} _ {t} $
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contains all subsets from  $  {\mathcal F} $
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of  $  {\mathsf P} $-
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measure zero.
  
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For stochastic bases, the notations  $  ( \Omega , {\mathcal F}, \mathbf F , {\mathsf P}) $
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or  $  ( \Omega , {\mathcal F} , ( {\mathcal F} _ {t} ) _ {t \geq  0 }  , {\mathsf P}) $
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are also used.
  
 
====Comments====
 
====Comments====
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An increasing family of (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090020/s09002013.png" />-) algebras is usually called a filtration.
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An increasing family of ( $  \sigma $-)  
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algebras is usually called a filtration.

Latest revision as of 08:23, 6 June 2020


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=48845
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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