When is the convex hull of a L\'evy path smooth?

TL;DR

This paper characterizes when the convex hull of a one-dimensional Lévy process is smooth, linking it to the process's transition laws and the behavior of its Lévy measure, especially at zero.

Contribution

It introduces the class of strongly eroded Lévy processes and characterizes smooth convex hulls in terms of the Lévy measure and process behavior, extending understanding of path regularity.

Findings

Broad class of infinite variation Lévy processes have smooth convex hulls.

Strongly eroded Lévy processes are exactly those with smooth convex hulls.

Conjecture that infinite variation Lévy processes are either strongly eroded or abrupt.

Abstract

We characterise, in terms of their transition laws, the class of one-dimensional L\'evy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is exhibited by a broad class of infinite variation L\'evy processes and depends subtly on the behaviour of the L\'evy measure at zero. We introduce a class of strongly eroded L\'evy 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 L\'evy 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 L\'evy measure.