Convex minorants and the fluctuation theory of Lévy processes

Abstract

We establish a novel characterisation of the law of the convex minorant of any Lévy process. Our self-contained elementary proof is based on the analysis of piecewise linear convex functions and requires only very basic properties of Lévy processes. Our main result provides a new simple and self-contained approach to the fluctuation theory of Lévy 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.

18 pages, 2 figures, updated references, new section 2.3 on the vertex process, short video on https://youtu.be/hEg4YmxOgXA

References in corpus (1)

• A Gaussian approximation theorem for Lévy processes

Cited by in corpus (3)

• Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation
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• Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function