martingale

Martingale is a term used in several domains, most notably probability theory, gambling, and equestrian tack. In probability theory, a martingale is a stochastic process that, at any time, has an expected future value equal to its current value given all past information. The term also appears in gambling contexts as a betting strategy and in horse riding as a type of head-control strap.

Formally, a martingale is defined on a filtered probability space (Ω, F, {F_t}, P). An adapted, integrable process {M_t} is a martingale if for all s ≤ t, E[M_t | F_s] = M_s. This expresses a fair-game condition: future expected value, given past information, equals the present value. Classic examples include the simple symmetric random walk and Brownian motion, both martingales with respect to their natural filtrations. Extensions include local martingales, which satisfy the condition up to a stopping time, and supermartingales, where E[M_t | F_s] ≤ M_s.

Gambling use: the martingale betting system. The strategy doubles the stake after every loss, aiming to recover all losses and gain a fixed amount on the next win. It can yield short-term gains but is risky and typically requires unlimited credit and no table limits; in games with a house edge, the expected value is negative and large losses are possible.

Horse tack: a martingale is a strap used on horses to limit head carriage by connecting the bridle to the girth or noseband. Standing martingales restrain upward movement, while running martingales include a movable piece for gradual control. They are used in training and performance riding.