Introduction to Martingales

TL;DR

This paper provides an introductory overview of Martingales, covering foundational measure theory, key theorems, and applications in finance, aimed at readers new to the topic.

Contribution

It offers a comprehensive introduction to Martingales, including proofs of fundamental theorems and their applications, filling a gap for learners in probability theory.

Findings

Proofs of Kolmogorov's Theorem and Martingale Property

Explanation of the Pythagorean Theorem on Martingales

Applications of Martingales in financial modeling

Abstract

This paper introduces Martingales by covering introductory measure theory concepts and the Lebesgue Integration and Conditional Expectation. It follows up with proofs of Kolomorgov's Theorem on conditional expectations, the Martingale Property, and the Pythagorean Theorem on Martingales. Finally, it ends with Martingales' applications in finance.