submartingales and supermartingales

Most real games are not perfectly fair. A casino's profit tends to creep upward in its favour; a careless gambler's fortune tends to bleed downward. To model these we relax the martingale's exact balance into an inequality and get two close cousins. A submartingale is a game that is, on average, favourable to the player — the forecast of the next value never falls below the present. A supermartingale is the opposite — on average unfavourable, with the forecast never rising above the present. They are the tilted versions of a fair game.

Precisely, with M_n integrable and adapted to the past, a submartingale satisfies E[M_(n+1) given the past) >= M_n for every n, while a supermartingale satisfies E[M_(n+1) given the past) <= M_n. A martingale is exactly the boundary case where both inequalities hold and you get equality. The names feel backwards at first, and there is a memory trick: 'super' sits high but tends to come DOWN (supermartingales decrease in expectation, like a falling fortune), while 'sub' sits low but tends to go UP. Taking expectations shows the mean of a submartingale is non-decreasing in n and the mean of a supermartingale is non-increasing. A handy fact: if M_n is a martingale and phi is a convex function (with phi(M_n) integrable), then phi(M_n) is a submartingale — for instance |M_n| and M_n^2 are submartingales, a fact that powers many inequalities.

These variants matter because nature rarely hands you an exactly fair process, and many martingale theorems extend to the tilted case with the equalities replaced by the matching inequalities — the optional stopping theorem, the maximal inequality and the convergence theorem all have submartingale and supermartingale forms. A core decomposition (Doob's) even says any submartingale splits uniquely into a martingale plus a predictable increasing 'drift', cleanly separating the fair part from the favourable trend.