The word "martingale" has long had associations with . Historically, a martingale was a strategy of doubling your stake every time you lose in order to recoup the loss. If you lose a dollar, two. If you lose again, four. Continuing with the strategy, you are almost guaranteed to finish with a dollar in total , but there is a remote chance that you will instead lose all you money. Today, "martingale" has a very different meaning. It is used in mathematics to describe a form of stochastic process that is akin to a fair .
Most people have an intuitive sense for what constitutes a fair . If you go to a and play , that is not a fair . On average the house wins money, so on average you lose money. A fair is one at which, on average, you break even.
Let be a stochastic process, and let denote a realization of at time t (see the notation conventions documentation). The process is a martingale if, at any time t, the expected value of all future values equals the current value . Formally, in the notation of conditional probabilities, this means
for all t and all .
If you are on a game of chance, the game is fair if and only if your wealth follows a martingale through playing. A martingale always has zero drift. For this reason, quantities such as equity or commodity prices are generally not modeled with martingales. Equity prices are generally assumed to drift upward with time while some commodity prices rise and fall in a seasonal manner. Nonetheless, martingales play a central role in financial engineering. Brownian motion and random walks are martingales, and martingales appear in the fundamental theorem of asset pricing.
If the second equality in  is replaced with , the process is a submartingale—a process with positive drift is a submartingale. If the second equality in  is replaced with , the process is a supermartingale—a process with negative drift, is a supermartingale. Note that neither inequality is strict, so martingales are included in both definitions.