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Difference between revisions of "Stochastic interval"

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One of the intervals:
 
One of the intervals:
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s0901401.png" /></td> </tr></table>
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$$
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\left [ \left [ \sigma , \tau \right ] \right ]  = \
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\{ {( \omega , t) } : {t \geq  0, \sigma ( \omega ) \leq  t
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\leq  \tau ( \omega ) } \} ,
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$$
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s0901402.png" /></td> </tr></table>
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$$
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\left [ \left [ \sigma , \tau \right [ \right [  = \
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\{ {( \omega , t) } : {t \geq  0,\
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\sigma ( \omega ) \leq  t < \tau ( \omega ) } \} ,
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$$
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s0901403.png" /></td> </tr></table>
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$$
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\left ] \left ] \sigma , \tau \right ] \right ]  = \
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\{ {( \omega , t) } : {t \geq  0,\
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\sigma ( \omega ) < t \leq  \tau ( \omega ) } \} ,
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$$
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s0901404.png" /></td> </tr></table>
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$$
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\left ] \left ] \sigma , \tau \right [ \right [  = \
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\{ {( \omega , t) } : {t \geq  0,\
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\sigma ( \omega ) < t < \tau ( \omega ) } \} ,
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$$
  
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s0901405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s0901406.png" /> are two stopping times defined on a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s0901407.png" /> with an increasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s0901408.png" /> of sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s0901409.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090140/s09014010.png" />.
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where $  \sigma = \sigma ( \omega ) $
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and $  \tau = \tau ( \omega ) $
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are two stopping times defined on a measurable space $  ( \Omega , {\mathcal F}) $
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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} $.
  
 
====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>

Latest revision as of 08:23, 6 June 2020


One of the intervals:

$$ \left [ \left [ \sigma , \tau \right ] \right ] = \ \{ {( \omega , t) } : {t \geq 0, \sigma ( \omega ) \leq t \leq \tau ( \omega ) } \} , $$

$$ \left [ \left [ \sigma , \tau \right [ \right [ = \ \{ {( \omega , t) } : {t \geq 0,\ \sigma ( \omega ) \leq t < \tau ( \omega ) } \} , $$

$$ \left ] \left ] \sigma , \tau \right ] \right ] = \ \{ {( \omega , t) } : {t \geq 0,\ \sigma ( \omega ) < t \leq \tau ( \omega ) } \} , $$

$$ \left ] \left ] \sigma , \tau \right [ \right [ = \ \{ {( \omega , t) } : {t \geq 0,\ \sigma ( \omega ) < t < \tau ( \omega ) } \} , $$

where $ \sigma = \sigma ( \omega ) $ and $ \tau = \tau ( \omega ) $ are two stopping times defined on a measurable space $ ( \Omega , {\mathcal F}) $ with an increasing family $ \mathbf F = ( {\mathcal F} _ {t} ) _ {t\geq } 0 $ of sub- $ \sigma $- algebras $ {\mathcal F} _ {t} \subseteq {\mathcal F} $.

References

[1] C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972)
How to Cite This Entry:
Stochastic interval. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_interval&oldid=48852
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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