Namespaces
Variants
Actions

Difference between revisions of "Predictable sigma-algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (wikilink)
m (tex encoded by computer)
 
Line 1: Line 1:
''predictable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p0744102.png" />-algebra''
+
<!--
 +
p0744102.png
 +
$#A+1 = 30 n = 0
 +
$#C+1 = 30 : ~/encyclopedia/old_files/data/P074/P.0704410 Predictable sigma\AAhalgebra,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
The least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p0744103.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p0744104.png" /> of sets in
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/p/p074/p074410/p0744105.png" /></td> </tr></table>
+
''predictable  $  \sigma $-
 +
algebra''
  
generated by all mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p0744106.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p0744107.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p0744108.png" /> that are (for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p0744109.png" />) continuous from the left (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441010.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441011.png" />-adapted to a non-decreasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441012.png" /> of sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441013.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441016.png" /> is a measurable space. A predictable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441017.png" />-algebra can be generated by any of the following families of sets:
+
The least  $  \sigma $-
 +
algebra  $  {\mathcal P} = {\mathcal P} ( \mathbf F ) $
 +
of sets in
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441021.png" /> is a stopping time (cf. [[Markov moment|Markov moment]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441022.png" /> a [[stochastic interval]];
+
$$
 +
\Omega \times \mathbf R _ {+}  = \
 +
\{ {( \omega , t) } : {\omega \in \Omega , t \geq  0 } \}
 +
$$
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441027.png" />.
+
generated by all mappings  $  ( \omega , t) \rightarrow f ( \omega , t) $
 +
of the set  $  \Omega \times \mathbf R _ {+} $
 +
into  $  \mathbf R $
 +
that are (for each fixed  $  \omega \in \Omega $)
 +
continuous from the left (in  $  t $)
 +
and  $  \mathbf F $-
 +
adapted to a non-decreasing family  $  \mathbf F = ( {\mathcal F} _ {t} ) _ {t \geq  0 }  $
 +
of sub- $  \sigma $-
 +
algebras  $  {\mathcal F} _ {t} \subseteq {\mathcal F} $,  
 +
$  t \geq  0 $,  
 +
where $  ( \Omega , {\mathcal F} ) $
 +
is a measurable space. A predictable  $  \sigma $-
 +
algebra can be generated by any of the following families of sets:
  
Between optional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441028.png" />-algebras (cf. [[Optional sigma-algebra|Optional sigma-algebra]]) and predictable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441029.png" />-algebras there is the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441030.png" />.
+
1)  $  A \times \{ 0 \} $,
 +
where  $  A \in {\mathcal F} _ {0} $
 +
and  $  [[ 0, \tau ]] $,
 +
where  $  \tau $
 +
is a stopping time (cf. [[Markov moment|Markov moment]]) and  $  [[ 0, \tau ]] $
 +
a [[stochastic interval]];
 +
 
 +
2)  $  A \times \{ 0 \} $,
 +
where  $  A \in {\mathcal F} _ {0} $,
 +
and  $  A \times ( s, t] $,
 +
where  $  s < t $
 +
and  $  A \in {\mathcal F} _ {s} $.
 +
 
 +
Between optional  $  \sigma $-
 +
algebras (cf. [[Optional sigma-algebra|Optional sigma-algebra]]) and predictable $  \sigma $-
 +
algebras there is the relation $  {\mathcal P} ( \mathbf F ) \subseteq {\mathcal O} ( \mathbf F ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Dellacherie,  "Capacités et processus stochastique" , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Dellacherie,  "Capacités et processus stochastique" , Springer  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Instead of  "(s-) algebra"  one more often uses (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074410/p07441031.png" />-) field.
+
Instead of  "(s-) algebra"  one more often uses ( $  \sigma $-)  
 +
field.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,  P.A. Meyer,  "Probabilities and potential" , '''A-C''' , North-Holland  (1978–1988)  (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-C''' , North-Holland  (1978–1988)  (Translated from French)</TD></TR></table>

Latest revision as of 08:07, 6 June 2020


predictable $ \sigma $- algebra

The least $ \sigma $- algebra $ {\mathcal P} = {\mathcal P} ( \mathbf F ) $ 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 $ that are (for each fixed $ \omega \in \Omega $) continuous from the left (in $ t $) and $ \mathbf F $- adapted to a non-decreasing family $ \mathbf F = ( {\mathcal F} _ {t} ) _ {t \geq 0 } $ of sub- $ \sigma $- algebras $ {\mathcal F} _ {t} \subseteq {\mathcal F} $, $ t \geq 0 $, where $ ( \Omega , {\mathcal F} ) $ is a measurable space. A predictable $ \sigma $- algebra can be generated by any of the following families of sets:

1) $ A \times \{ 0 \} $, where $ A \in {\mathcal F} _ {0} $ and $ [[ 0, \tau ]] $, where $ \tau $ is a stopping time (cf. Markov moment) and $ [[ 0, \tau ]] $ a stochastic interval;

2) $ A \times \{ 0 \} $, where $ A \in {\mathcal F} _ {0} $, and $ A \times ( s, t] $, where $ s < t $ and $ A \in {\mathcal F} _ {s} $.

Between optional $ \sigma $- algebras (cf. Optional sigma-algebra) and predictable $ \sigma $- algebras there is the relation $ {\mathcal P} ( \mathbf F ) \subseteq {\mathcal O} ( \mathbf F ) $.

References

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

Comments

Instead of "(s-) algebra" one more often uses ( $ \sigma $-) field.

References

[a1] C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A-C , North-Holland (1978–1988) (Translated from French)
How to Cite This Entry:
Predictable sigma-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Predictable_sigma-algebra&oldid=48279
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article