Theory of Martingales

In this article, we discuss some advanced results in the theory of martingales and their application in operations research and management science. These results include (i) martingale inequalities, (ii) martingale limit theorem, and (iii) the optional stopping theorem. We also present a brief discussion on martingales with continuous parameter. This is a continuation of the articles Discrete-Time Martingales and Continuous-Time Martingales. Keywords: martingale; martingale inequalities; martingale convergence theorem; optional stopping; Brownian motion