Two continua of embedded regenerative sets

Abstract

Given a two-sided real-valued Lévy 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 $M_t:=\sup \{h\in \mathbb{R}:h-\alpha |t-s|\le X_s\text{ for all }s\in \mathbb{R}\}=\inf \{X_s+\alpha |t-s|:s\in \mathbb{R}\}$, $t\in \mathbb{R}$. 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 càdlàg, 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 $\beta$.

11 pages