Lipschitz minorants of Brownian Motion and 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 $m\le f$ 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 Lévy 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 Lévy measure of the associated subordinator. We study the limit of $\mathcal{Z}$ as $\alpha \to \infty$ and find for the so-called abrupt Lévy 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.

42 pages, 3 figures, revised to incorporate comments from readers plus further results on the behavior of Levy processes at their local extrema and extra references

References in corpus (2)

• The convex minorant of a Lévy process
• Convex minorants of random walks and Lévy processes