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''optional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o0685702.png" />-algebra''
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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]]).
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''optional  -
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algebra''
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The smallest   \sigma -
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algebra $  {\mathcal O} = {\mathcal O} ( \mathbf F ) $
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of sets (cf. [[Algebra of sets|Algebra of sets]]) in $  \Omega \times \mathbf R _ {+} = \{ {( \omega , t) } : {\omega \in \Omega,  t \geq  0 } \} $
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generated by all mappings   ( \omega , t) \rightarrow f( \omega , t)
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of the set   \Omega \times \mathbf R _ {+}
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into   \mathbf R
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which (for every fixed   \omega \in \Omega )  
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are continuous from the right (in   t ),  
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have limits from the left and are adapted to a (given) non-decreasing family $  \mathbf F = ( F _ {t} ) _ {t \geq  0 }  $
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of sub-   \sigma -
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algebras   F _ {t} \subseteq F ,  
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$  t \geq  0 $,  
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where   ( \Omega , F  )
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is a measurable space. The optional   \sigma -
 +
algebra coincides with the smallest   \sigma -
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algebra generated by the [[stochastic interval]]s $  [ 0, \tau ] = \{ {( \omega , t) } : {0 \leq  t < \tau ( \omega ) } \} $,  
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where $  \tau = \tau ( \omega ) $
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are stopping times (relative to $  \mathbf F = ( F _ {t} ) _ {t \geq  0 }  $)  
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(cf. [[Markov moment|Markov moment]]). The inclusion   {\mathcal P} ( \mathbf F ) \subseteq {\mathcal O} ( \mathbf F )
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holds between the optional and predictable   \sigma -
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algebras (cf. [[Predictable sigma-algebra|Predictable sigma-algebra]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Dellacherie,  "Capacités et processus stochastiques" , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Dellacherie,  "Capacités et processus stochastiques" , Springer  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
In [[#References|[a1]]] the optional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857023.png" />-field is called the well-measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068570/o06857025.png" />-field.
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In [[#References|[a1]]] the optional   \sigma -
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field is called the well-measurable   \sigma -
 +
field.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,  P.A. Meyer,  "Probabilities and potential" , '''A''' , North-Holland  (1978)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,  P.A. Meyer,  "Probabilities and potential" , '''A''' , North-Holland  (1978)  (Translated from French)</TD></TR></table>

Latest revision as of 08:04, 6 June 2020


optional \sigma - algebra

The smallest \sigma - algebra {\mathcal O} = {\mathcal O} ( \mathbf F ) of sets (cf. Algebra of sets) in \Omega \times \mathbf R _ {+} = \{ {( \omega , t) } : {\omega \in \Omega, t \geq 0 } \} generated by all mappings ( \omega , t) \rightarrow f( \omega , t) of the set \Omega \times \mathbf R _ {+} into \mathbf R which (for every fixed \omega \in \Omega ) are continuous from the right (in t ), have limits from the left and are adapted to a (given) non-decreasing family \mathbf F = ( F _ {t} ) _ {t \geq 0 } of sub- \sigma - algebras F _ {t} \subseteq F , t \geq 0 , where ( \Omega , F ) is a measurable space. The optional \sigma - algebra coincides with the smallest \sigma - algebra generated by the stochastic intervals [ 0, \tau ] = \{ {( \omega , t) } : {0 \leq t < \tau ( \omega ) } \} , where \tau = \tau ( \omega ) are stopping times (relative to \mathbf F = ( F _ {t} ) _ {t \geq 0 } ) (cf. Markov moment). The inclusion {\mathcal P} ( \mathbf F ) \subseteq {\mathcal O} ( \mathbf F ) holds between the optional and predictable \sigma - algebras (cf. Predictable sigma-algebra).

References

[1] C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972)

Comments

In [a1] the optional \sigma - field is called the well-measurable \sigma - 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
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article