The Levy-Ito Decomposition theorem

TL;DR

This paper provides an accessible, martingale-based proof of the Levy-Ito Decomposition theorem, clarifying the structure of infinitely divisible processes for students, especially in financial mathematics.

Contribution

It offers a pedagogically improved, detailed translation and explanation of Le9vy's decomposition theorem, emphasizing martingale methods.

Findings

Levy-Khintchine representation derived as a by-product

Martingale methods used for proof, enhancing pedagogical clarity

Accessible explanation suitable for students in financial mathematics

Abstract

This a free translation with additional explanations of {\em Processus \`a Accroissement Independants Chapitre I: La D\'ecomposition de Paul L\'evy}, by J.L. Bretagnolle, in {\em Ecole d'Et\'e de Probabilit\'es}, Lecture Notes in Mathematics 307, Springer 1973. The L\'evy-Khintchine representation of infinitely divisible distributions is obtained as a by-product. As this proof makes use of martingale methods, it is pedagogically more suitable for students of financial mathematics than some other approaches. It is hoped that the end notes will also help to make the proof more accessible to this audience.