A martingale is the mathematical description of a fair game: The expected net gain or loss from further play, independent of the history, is 0. The rigorous mathematical definition involves conditional expectation. Following earlier work by Paul Lévy and Jean Ville, Joseph Leo Doob developed in the 1940s and 1950s basic martingale theory and many of its applications. Martingales allowed one to study, for the first time, the behavior of sums and sequences of random variables which are not independent. Martingale theory is one of the cornerstones of modern mathematical probability theory with wide-ranging applications in stochastic analysis and mathematical finance.