Given a two-sided real-valued Levy process $(X_t)_{t in mathbb{R}}$, define processes $(L_t)_{t in mathbb{R}}$ and $(M_t)_{t in mathbb{R}}$ by $L_t := sup{h in mathbb{R} : h - alpha(t-s) le X_s text{ for all } s le t} = inf{X_s + alpha(t-s) : s le t}$, $t in mathbb{R}$, and $M_t := sup { h in mathbb{R} : h - alpha|t-s| leq X_s text{ for all } s in mathbb{R} } = inf {X_s + alpha |t-s| : s in mathbb{R}}$, $t in mathbb{R}$. The corresponding contact sets are the random sets $mathcal{H}_alpha := { t in mathbb{R} : X_{t}wedge X_{t-} = L_t}$ and $mathcal{Z}_alpha := { t in mathbb{R} : X_{t}wedge X_{t-} = M_t}$. For a fixed $alpha>mathbb{E}[X_1]$ (resp. $alpha>|mathbb{E}[X_1]|$) the set $mathcal{H}_alpha$ (resp. $mathcal{Z}_alpha$) is non-empty, closed, unbounded above and below, stationary, and regenerative. The collections $(mathcal{H}_{alpha})_{alpha > mathbb{E}[X_1]}$ and $(mathcal{Z}_{alpha})_{alpha > |mathbb{E}[X_1]|}$ are increasing in $alpha$ and the regeneration property is compatible with these inclusions in that each family is a continuum of embedded regenerative sets in the sense of Bertoin. We show that $(sup{t < 0 : t in mathcal{H}_alpha})_{alpha > mathbb{E}[X_1]}$ is a c`adl`ag, nondecreasing, pure jump process with independent increments and determine the intensity measure of the associated Poisson process of jumps. We obtain a similar result for $(sup{t < 0 : t in mathcal{Z}_alpha})_{alpha > |beta|}$ when $(X_t)_{t in mathbb{R}}$ is a (two-sided) Brownian motion with drift $beta$.
Download