A short Proof of the Doob-Meyer Theorem

TL;DR

This paper presents a concise and elementary proof of the Doob-Meyer decomposition theorem, which states that every submartingale of class D can be uniquely decomposed into a martingale and a predictable increasing process.

Contribution

It offers a simplified, self-contained proof of the Doob-Meyer theorem, including multiple known arguments for clarity.

Findings

Provides a short, elementary proof of the Doob-Meyer decomposition

Ensures the proof is self-contained with multiple known arguments

Confirms the uniqueness of the decomposition for submartingales of class D

Abstract

Every submartingale S of class D has a unique Doob-Meyer decomposition S=M+A, where M is a martingale and A is a predictable increasing process starting at 0. We provide a short and elementary prove of the Doob-Meyer decomposition theorem. Several previously known arguments are included to keep the paper self-contained.