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Difference between revisions of "Stochastic indistinguishability"

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[[Category:Stochastic processes]]
 
[[Category:Stochastic processes]]
  
A property of two random processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901202.png" /> which states that the random set
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A property of two random processes $  X = ( X _ {t} ( \omega )) _ {t \geq  0 }  $
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and $  Y = ( Y _ {t} ( \omega )) _ {t \geq  0 }  $
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which states that the random set
  
<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/s090120/s0901203.png" /></td> </tr></table>
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$$
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\{ X \neq Y \}  = \
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\{ {( \omega , t) } : {X _ {t} ( \omega ) \neq Y _ {t} ( \omega ) } \}
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$$
  
can be disregarded, i.e. that the probability of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901204.png" /> is equal to zero. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901206.png" /> are stochastically indistinguishable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901207.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901208.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s09012010.png" /> are stochastically equivalent (cf. [[Stochastic equivalence|Stochastic equivalence]]). The opposite, generally speaking, is not true, but for processes that are continuous from the right (left), stochastic indistinguishability follows from stochastic equivalence.
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can be disregarded, i.e. that the probability of the set $  \{  \omega  : {\exists t \geq  0 : ( \omega , t) \in \{ X \neq Y \} } \} $
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is equal to zero. If $  X $
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and $  Y $
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are stochastically indistinguishable, then $  X _ {t} = Y _ {t} $
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for all $  t \geq  0 $,  
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i.e. $  X $
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and $  Y $
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are stochastically equivalent (cf. [[Stochastic equivalence|Stochastic equivalence]]). The opposite, generally speaking, is not true, but for processes that are continuous from the right (left), stochastic indistinguishability follows from stochastic equivalence.
  
 
====References====
 
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====Comments====
 
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====References====

Latest revision as of 08:23, 6 June 2020


2020 Mathematics Subject Classification: Primary: 60Gxx [MSN][ZBL]

A property of two random processes $ X = ( X _ {t} ( \omega )) _ {t \geq 0 } $ and $ Y = ( Y _ {t} ( \omega )) _ {t \geq 0 } $ which states that the random set

$$ \{ X \neq Y \} = \ \{ {( \omega , t) } : {X _ {t} ( \omega ) \neq Y _ {t} ( \omega ) } \} $$

can be disregarded, i.e. that the probability of the set $ \{ \omega : {\exists t \geq 0 : ( \omega , t) \in \{ X \neq Y \} } \} $ is equal to zero. If $ X $ and $ Y $ are stochastically indistinguishable, then $ X _ {t} = Y _ {t} $ for all $ t \geq 0 $, i.e. $ X $ and $ Y $ are stochastically equivalent (cf. Stochastic equivalence). The opposite, generally speaking, is not true, but for processes that are continuous from the right (left), stochastic indistinguishability follows from stochastic equivalence.

References

[D] C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) MR0448504 Zbl 0246.60032

Comments

References

[DM] C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A , North-Holland (1978) (Translated from French) MR0521810 Zbl 0494.60001
How to Cite This Entry:
Stochastic indistinguishability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_indistinguishability&oldid=48850
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article