Half-martingale

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.