ABSTRACT
This chapter presents the major results of classical martingale theory, such as maximal inequality, upcrossing inequality, Doob’s inequality, convergence theorems, Riesz decomposition theorem, and Doob’s stopping theorem. It deals the discrete time case and the continuous time case. In order to deepen readers’ understanding, the chapter illustrates some examples of applications at times. It gives an introduction to processes with independent increments. In order to establish the elementary inequalities of martingales and supermartingales the authors have proved Doob’s stopping theorem (or so-called optional sampling theorem) for bounded stopping times. There are two kinds of generalizations. The first is to a class of closable martingales and supermartingales. The second is to general martingales and supermartingales.
