2011/10/05 by Joshua Abramson, Steven N. Evans, Abramson, Joshua +1
Mathematics · #60G17 #60G51 #60G55 #60J65 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60G17 #msc:60G51 #msc:60G55 #msc:60J65
paper · pdf · doi:10.48550/arxiv.1110.1105
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
arxiv created 2012/03/03 · arxiv updated 2012/03/06
For α> 0, the α-Lipschitz minorant of a function f: ℝ → ℝ is the greatest function m : ℝ → ℝ such that m ≤ f and |m(s)-m(t)| ≤ α|s-t| for all s,t ∈ ℝ, should such a function exist. If X=(Xt)t ∈ ℝ is a real-valued Lévy process that is not pure linear drift with slope ± α, then the sample paths of X have an α-Lipschitz minorant almost surely if and only if | 𝔼[X1] | < α. Denoting the minorant by M, we investigate properties of the random closed set Z := t ∈ ℝ : Mt = Xt \wedge Xt-, 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 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 Z as α→ ∞ 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 β such that |β| < α, we calculate explicitly the densities of various random variables related to the minorant.