Basics of Martingales
This chapter was a brief introduction to the martingales. In the first part of the chapter; the definitions of a martingale, submartingale, supermartingale, and Doob type martingale were provided, and some examples were given to show how to verify whether a stochastic process is a martingale, submartingale or supermartingale. In the second part of the chapter, Azuma-Hoeffding inequality, Kolmogorov’s inequality for submartingales with an extension to martingales, and the martingale convergence theorem have been defined as the selected theorems of the martingales. Two problems were presented to show that a standard Brownian motion and a special type of Poisson process are martingales.