A Martingale* is a stochastic process with sequential variables Xi (i=1,2,..) where the expected value of each variable is the same and independent of previous observations:

E\big(X_{n+1}\big)=E\big(X_{n+1}\mid X_1,\dots , X_n\big)=E\big(X_n\big) |

Thus a Martingale is any stochastic process with a constant mean. The theory was originally developed to demonstrate the fairness of gambling games, i.e. to show that the expected winnings of each turn of a game is constant: for example, that remembering the cards that have already been played in previous hands of a card game wouldn't impact your expected winnings. However the theory has proven to be of considerable value in many real world problems.

*A Martingale gets its name from the gambling "system' of doubling your bet on each loss of an even odds bet (e.g. betting Red or Impaire at the roulette wheel) until you have a win. It works too – well, in theory anyway. You must have a huge bankroll, and the casino must have no bet limit. It gives low returns for high risk, so as risk analysis consultants we would advise you to invest in (gamble on) the stock market instead.