The convex minorant of a Lévy process

Abstract

We offer a unified approach to the theory of convex minorants of Lévy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a Lévy 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 Lévy process with continuous distributions above its convex minorant on the interval $[0,1]$.

Published in at http://dx.doi.org/10.1214/11-AOP658 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Bridges of Lévy processes conditioned to stay positive
• Convex minorants of random walks and Lévy processes
• On the law of the supremum of Lévy processes
• The greatest convex minorant of Brownian motion, meander, and bridge

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