Foundation of Probability Theory and Discrete-Time Martingales
Gambling with dice was very popular in medieval Europe. The study of problems involving probability associated with gambling led to the development of probability theory. However, it was not until the early twentieth century that probability theory was considered as a branch of mathematics. The mathematical foundation of modern probability theory was laid by Andrei N. Kolmogorov in 1933. He adopted Lebesgue’s framework of measure theory and created an axiomatic system for probability theory. This chapter introduces some basic concepts and results of modern probability theory, highlights the results related to the conditional mathematical expectation, and then introduces discrete-time martingale theory, including the martingale transform and the Snell envelope. We assume that the reader has basic knowledge of measure theory.