ABSTRACT
The definition of a martingale as given in this chapter heavily relies on the concept of the conditional mean value of a random variable given values of other random variables or on the concept of the conditional mean value of a random variable given other random variables. Martingales are important tools for solving prestigious problems in probability theory and its applications. Heuristically, martingales are stochastic models for 'fair games' in a wider sense, i.e., games, in which each side has the same chance to win or to lose. Martingales were introduced as a special class of stochastic processes by J. Ville und P. Levy. Analogously to Markov processes, the terminology discrete-time martingales and continuous-time martingales is adopted. Most of the literature on martingales is measure-theoretically based. In this case, the definition of a martingale is usually done by means of the concept of a filtration.
