Convex minorants and the fluctuation theory of L\'evy processes

TL;DR

This paper introduces a new, elementary characterization of the convex minorant of any Lévy process, simplifying the fluctuation theory and deriving classical results through basic convex analysis.

Contribution

It provides a novel, self-contained approach to Lévy process fluctuation theory, avoiding complex tools like local time and excursion theory.

Findings

New characterization of the convex minorant law

Simplified proof of fluctuation theory results

Derivation of classical theorems and a new factorization identity

Abstract

We establish a novel characterisation of the law of the convex minorant of any L\'evy process. Our self-contained elementary proof is based on the analysis of piecewise linear convex functions and requires only very basic properties of L\'evy processes. Our main result provides a new simple and self-contained approach to the fluctuation theory of L\'evy processes, circumventing local time and excursion theory. Easy corollaries include classical theorems, such as Rogozin's regularity criterion, Spitzer's identities and the Wiener-Hopf factorisation, as well as a novel factorisation identity.