ABSTRACT
In this chapter, we will learn the definition, some basic facts and the optional sampling theorem for discrete-time martingales, as well as three kinds of useful inequalities (Lenglart's, Bernstein's, and Burkholder's inequalities). All of them will be generalized to the ones for the continuous-time case in subsequent chapters. Among the contents in this chapter, an inconspicuous but important subject is the martingale transformation. Although it may look trivial or less interesting at first sight, it is a very important fact which gives the prototypes for the stochastic integrals and the predictable quadratic variation in the theory of continuous-time martingales.
