A proof of the continuous martingale convergence theorem

TL;DR

This paper presents a proof of the continuous martingale convergence theorem using classical inequalities and properties of super-martingales, providing a rigorous foundation for understanding martingale behavior.

Contribution

It offers a new proof of the continuous martingale convergence theorem based on classical inequalities and super-martingale properties.

Findings

Proof relies on classical martingale inequalities.

Uses truncation argument for super-martingale convergence.

Establishes almost sure convergence of martingales.

Abstract

A proof of the continuous martingale convergence theorem is provided. It relies on a classical martingale inequality and the almost sure convergence of a uniformly bounded non-negative super-martingale, after a truncation argument.