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

Jorge Ignacio González Cázares; Aleksandar Mijatović

doi:10.30757/alea.v19-39

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arXiv · 2105.15060

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

Jorge Ignacio González Cázares·Aleksandar Mijatović

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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.

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Jorge Ignacio González Cázares, Aleksandar Mijatović. 2021-05-31. Convex minorants and the fluctuation theory of L\'evy processes. https://doi.org/10.30757/alea.v19-39

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