Difference between revisions of "Half-martingale"
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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} ) $, | ||
| + | $ 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 | 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 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.
Half-martingale. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Half-martingale&oldid=13853