An Introduction to the Modern Martingale Theory and Applications: An Analytic View: 81 (Texts in Applied Mathematics, 81)

Martingale theory is a cornerstone of modern probability, offering a natural extension of the study of sums of independent random variables.Although its roots can be traced back to the work of Paul Lévy in 1937, it was Joseph L.Doob in the 1940s who formally developed the theory, culminating in his landmark book Stochastic Processes in 1953.Since then, martingale theory has evolved significantly, with deep contributions from mathematicians such as Donald L.Burkholder, Richard Gundy, and Burgess Davis, among others.This is what is now known as advanced martingale theory, which began with the publication of Burkholder’s seminal paper Martingale Transforms in 1966. This book provides a comprehensive treatment of both classical and advanced martingale theory.It opens with a historical introduction, exploring foundational functions such as Rademacher, Haar, and Walsh functions, before delving into the core concepts of conditional probability.The classical theory, as developed by Doob, is me