Difference between revisions of "Optional sigma-algebra"
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''optional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685702.png" />-algebra'' | ''optional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685702.png" />-algebra'' | ||
| − | The smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685703.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685704.png" /> of sets (cf. [[Algebra of sets|Algebra of sets]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685705.png" /> generated by all mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685706.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685707.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685708.png" /> which (for every fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685709.png" />) are continuous from the right (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857010.png" />), have limits from the left and are adapted to a (given) non-decreasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857011.png" /> of sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857012.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857015.png" /> is a measurable space. The optional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857016.png" />-algebra coincides with the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857017.png" />-algebra generated by the stochastic | + | The smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685703.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685704.png" /> of sets (cf. [[Algebra of sets|Algebra of sets]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685705.png" /> generated by all mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685706.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685707.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685708.png" /> which (for every fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685709.png" />) are continuous from the right (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857010.png" />), have limits from the left and are adapted to a (given) non-decreasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857011.png" /> of sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857012.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857015.png" /> is a measurable space. The optional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857016.png" />-algebra coincides with the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857017.png" />-algebra generated by the [[stochastic interval]]s <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857019.png" /> are stopping times (relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857020.png" />) (cf. [[Markov moment|Markov moment]]). The inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857021.png" /> holds between the optional and predictable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857022.png" />-algebras (cf. [[Predictable sigma-algebra|Predictable sigma-algebra]]). |
====References==== | ====References==== | ||
Revision as of 06:36, 2 October 2016
optional 
-algebra
The smallest 
-algebra 
of sets (cf. Algebra of sets) in 
generated by all mappings 
of the set 
into 
which (for every fixed 
) are continuous from the right (in 
), have limits from the left and are adapted to a (given) non-decreasing family 
of sub-
-algebras 
, 
, where 
is a measurable space. The optional 
-algebra coincides with the smallest 
-algebra generated by the stochastic intervals 
, where 
are stopping times (relative to 
) (cf. Markov moment). The inclusion 
holds between the optional and predictable 
-algebras (cf. Predictable sigma-algebra).
References
| [1] | C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) |
Comments
In [a1] the optional 
-field is called the well-measurable 
-field.
References
| [a1] | C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A , North-Holland (1978) (Translated from French) |
How to Cite This Entry:
Optional sigma-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Optional_sigma-algebra&oldid=39354
Optional sigma-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Optional_sigma-algebra&oldid=39354
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article