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When is the convex hull of a Lévy path smooth?

David Bang (University of Warwick)
Tuesday 15 November 2022 13:15–14:00 Koll. G3 (1532-218)
Stochastics Seminar

Abstract: We characterise, in terms of their transition laws, the class of one-dimensional Lévy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is exhibited by a broad class of infinite variation Lévy processes and depends subtly on the behaviour of the Lévy measure at zero. We introduce a class of strongly eroded Lévy processes, whose Dini derivatives vanish at every local minimum of the trajectory for all perturbations with a linear drift, and prove that these are precisely the processes with smooth convex hulls. We study how the smoothness of the convex hull can break and construct examples exhibiting a variety of smooth/non-smooth behaviours. Finally, we conjecture that an infinite variation Lévy process is either strongly eroded or abrupt, a claim implied by Vigon’s point-hitting conjecture. In the finite variation case, we characterise the points of smoothness of the hull in terms of the Lévy measure.

Link to paper: https://arxiv.org/abs/2205.14416

Organised by: Stochastics Group
Contact: Andreas Basse-O'Connor Revised: 07.11.2022