The convex minorant of a L\'{e}vy process

TL;DR

This paper develops a unified theory for convex minorants of Lévy processes with continuous distributions, providing explicit constructions and revealing the Poisson-Dirichlet distribution as a universal law for excursion lengths.

Contribution

It introduces explicit constructions of convex minorants for Lévy processes and characterizes the distribution of excursion lengths as Poisson-Dirichlet, unifying various aspects of the theory.

Findings

Explicit constructions of convex minorants on finite and infinite intervals

Poisson point process of excursions characterized

Poisson-Dirichlet distribution as universal law for excursion lengths

Abstract

We offer a unified approach to the theory of convex minorants of L\'{e}vy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a L\'{e}vy process on both finite and infinite time intervals, and of a Poisson point process of excursions above the convex minorant up to an independent exponential time. The Poisson-Dirichlet distribution of parameter 1 is shown to be the universal law of ranked lengths of excursions of a L\'{e}vy process with continuous distributions above its convex minorant on the interval $[0,1]$.