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A concept equivalent to either the concept of a submartingale or that of a supermartingale. A stochastic sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h0461601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h0461602.png" />, defined on a probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h0461603.png" /> with a distinguished non-decreasing family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h0461604.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h0461605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h0461606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h0461607.png" />, is called a half-martingale if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h0461608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h0461609.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046160/h04616010.png" />-measurable and with probability 1 either
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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/h/h046/h046160/h04616011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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A concept equivalent to either the concept of a submartingale or that of a supermartingale. A stochastic sequence  $  X = ( X _ {t} , {\mathcal F} _ {t} ) $,
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$  t \in T \subseteq [ 0 , \infty ) $,
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defined on a probability space  $  ( \Omega , {\mathcal F} , {\mathsf P}) $
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with a distinguished non-decreasing family of  $  \sigma $-
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algebras  $  ( {\mathcal F} _ {t} ) _ {t \in T }  $,
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$  {\mathcal F} _ {s} \subseteq {\mathcal F} _ {t} \subseteq {\mathcal F} $,
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$  s \leq  t $,
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is called a half-martingale if  $  {\mathsf E} | X _ {t} | < \infty $,
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$  X _ {t} $
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is  $  {\mathcal F} _ {t} $-
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measurable and with probability 1 either
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$$ \tag{1 }
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{\mathsf E} ( X _ {t}  \mid  {\mathcal F} _ {s} ) \geq  X _ {s} ,
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$$
  
 
or
 
or
  
<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/h/h046/h046160/h04616012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
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{\mathsf E} ( X _ {t}  \mid  {\mathcal F} _ {s} ) \leq  X _ {s} .
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$$
  
 
In case (1) the sequence is called a submartingale, and in case (2) — a supermartingale.
 
In case (1) the sequence is called a submartingale, and in case (2) — a supermartingale.
  
 
In the modern literature, the term  "half-martingale"  is either not used at all or identified with the concept of a submartingale (supermartingales are derived from submartingales by a change of sign and are sometimes called lower half-martingales). See also [[Martingale|Martingale]].
 
In the modern literature, the term  "half-martingale"  is either not used at all or identified with the concept of a submartingale (supermartingales are derived from submartingales by a change of sign and are sometimes called lower half-martingales). See also [[Martingale|Martingale]].

Latest revision as of 19:42, 5 June 2020


A concept equivalent to either the concept of a submartingale or that of a supermartingale. A stochastic sequence $ X = ( X _ {t} , {\mathcal F} _ {t} ) $, $ t \in T \subseteq [ 0 , \infty ) $, defined on a probability space $ ( \Omega , {\mathcal F} , {\mathsf P}) $ with a distinguished non-decreasing family of $ \sigma $- algebras $ ( {\mathcal F} _ {t} ) _ {t \in T } $, $ {\mathcal F} _ {s} \subseteq {\mathcal F} _ {t} \subseteq {\mathcal F} $, $ s \leq t $, is called a half-martingale if $ {\mathsf E} | X _ {t} | < \infty $, $ X _ {t} $ is $ {\mathcal F} _ {t} $- measurable and with probability 1 either

$$ \tag{1 } {\mathsf E} ( X _ {t} \mid {\mathcal F} _ {s} ) \geq X _ {s} , $$

or

$$ \tag{2 } {\mathsf E} ( X _ {t} \mid {\mathcal F} _ {s} ) \leq X _ {s} . $$

In case (1) the sequence is called a submartingale, and in case (2) — a supermartingale.

In the modern literature, the term "half-martingale" is either not used at all or identified with the concept of a submartingale (supermartingales are derived from submartingales by a change of sign and are sometimes called lower half-martingales). See also Martingale.

How to Cite This Entry:
Half-martingale. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Half-martingale&oldid=47161
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article