Probability and Stochastic Processes

Chapter 14Martingales

A martingale originally referred to a betting strategy. In eighteenth century France, especially due to Monaco's Monte Carlo and its casinos, it was not unusual for gamblers with a sure winning strategy (a martingale) to speak with probabilists and ask their opinion. The word itself was first mentioned in the thesis of Jean Ville (Ville, 1939), where it refers to a “gaming strategy” or martingale (for an entire etymology of the word, see Mansuy (2009)). Joseph Doob, who wrote a classic Doob (1953) in which martingales were established as one of the important groups of stochastic processes, later explains (Snell, 1997) that after reviewing the book by Ville in 1939 he was intrigued by the work and after formalizing the definition he kept the name.

Martingales may be defined as processes in continuous time or in discrete time. In this chapter, we consider only discrete-time martingales. In general, continuous-time martingales have the same basic properties as the discrete-time ones. The differences between discrete-time and continuous-time martingales come from analyzing the situation when two times s and t are very close to each other. We shall talk more about the differences born out of this issue later in this chapter. For a complete and wonderful reference, consult Karatzas and Shreve (1991), and for the notions of local martingales and semimartingales, look at Protter (2003).

14.1 Definition and Examples

Definition 14.1 (Discrete-time martingale) ...