A concept equivalent to either the concept of a submartingale or that of a supermartingale. A stochastic sequence , , defined on a probability space with a distinguished non-decreasing family of -algebras , , , is called a half-martingale if , is -measurable and with probability 1 either

(1)

or

(2)

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.