Lipschitz minorants of Brownian Motion and Levy processes

TL;DR

This paper studies the properties of the lpha Lipschitz minorant of Levy processes, characterizing the contact set, its distribution, and limits, with explicit results for Brownian motion with drift.

Contribution

It provides new conditions for the existence of the Lipschitz minorant, characterizes the contact set as a stationary regenerative process, and explores limits for abrupt Levy processes.

Findings

Contact set is countable or has zero Lebesgue measure under certain conditions.

Explicit formulas for the Levy measure of the associated subordinator.

Limit of contact set as lpha   the set of local infima for abrupt Levy processes.

Abstract

For $\alpha > 0$, the $\alpha$-Lipschitz minorant of a function $f: \mathbb{R} \to \mathbb{R}$ is the greatest function $m : \mathbb{R} \to \mathbb{R}$ such that $m \leq f$ and $|m(s)-m(t)| \le \alpha |s-t|$ for all $s,t \in \mathbb{R}$, should such a function exist. If $X=(X_t)_{t \in \mathbb{R}}$ is a real-valued L\'evy process that is not pure linear drift with slope $\pm \alpha$, then the sample paths of $X$ have an $\alpha$-Lipschitz minorant almost surely if and only if $| \mathbb{E}[X_1] | < \alpha$. Denoting the minorant by $M$, we investigate properties of the random closed set $\mathcal{Z} := {t \in \mathbb{R} : M_t = X_t \wedge X_{t-}}$, which, since it is regenerative and stationary, has the distribution of the closed range of some subordinator "made stationary" in a suitable sense. We give conditions for the contact set $\mathcal{Z}$ to be countable or to have zero Lebesgue measure, and we obtain formulas that characterize the L\'evy measure of the associated subordinator. We study the limit of $\mathcal{Z}$ as $\alpha \to \infty$ and find for the so-called abrupt L\'evy processes introduced by Vigon that this limit is the set of local infima of $X$. When $X$ is a Brownian motion with drift $\beta$ such that $|\beta| < \alpha$, we calculate explicitly the densities of various random variables related to the minorant.