论文标题
两个嵌入式再生集的连续性
Two continua of embedded regenerative sets
论文作者
论文摘要
给定一个双面实价的lévy过程$（x_t）_ {t \ in \ mathbb {r}} $，定义过程$（l_t）_ {t \ in \ mathbb {r}}}} $ and $（m_t）_ {m_t）_ {t \ in \ in \ mathbb} \ in \ Mathbb {r}：H - α（t-s）\ le x_s \ text {用于所有} s \ le t \} = \ inf \ {x_s + {x_s +α（t-s）：s \ le t \ \} \ Mathbb {R}：H -α| T -S | \ leq x_s \ text {for All} s \ in \ mathbb {r} \} = \ inf \ {x_s +α| t-s | ：s \ in \ mathbb {r} \} $，$ t \ in \ mathbb {r} $。相应的联系人集是随机集$ \ Mathcal {h}_α：= \ {t \ in \ Mathbb {r}：x_ {t} \ wedge x_ {t-} = l_t} = l_t \} $ x_ {t} \ wedge x_ {t-} = m_t \} $。对于固定的$α> \ mathbb {e} [x_1] $（resp。$α> | \ mathbb {e} [x_1] | $）set $ \ mathcal {h}_α$（seversive and indernary and in and insentery，indernary，indernary，indernary，indernary，indernary，indernary，indernary，inderaviencation The collections $(\mathcal{H}_α)_{α> \mathbb{E}[X_1]}$ and $(\mathcal{Z}_α)_{α> |\mathbb{E}[X_1]|}$ are increasing in $α$ and the regeneration property is compatible with these inclusions in that each family is a嵌入式再生集的连续体从贝托因（Bertoin）的意义上讲。 We show that $(\sup\{t < 0 : t \in \mathcal{H}_α\})_{α> \mathbb{E}[X_1]}$ is a càdlàg, nondecreasing, pure jump process with independent increments and determine the intensity measure of the associated Poisson process of jumps.我们获得了$（\ sup \ {t <0：t \ in \ mathcal {z}_α\}）_ {α> |β|} $时的类似结果。
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 $L_t := \sup\{h \in \mathbb{R} : h - α(t-s) \le X_s \text{ for all } s \le t\} = \inf\{X_s + α(t-s) : s \le t\}$, $t \in \mathbb{R}$, and $M_t := \sup \{ h \in \mathbb{R} : h - α|t-s| \leq X_s \text{ for all } s \in \mathbb{R} \} = \inf \{X_s + α|t-s| : s \in \mathbb{R}\}$, $t \in \mathbb{R}$. The corresponding contact sets are the random sets $\mathcal{H}_α:= \{ t \in \mathbb{R} : X_{t}\wedge X_{t-} = L_t\}$ and $\mathcal{Z}_α:= \{ t \in \mathbb{R} : X_{t}\wedge X_{t-} = M_t\}$. For a fixed $α>\mathbb{E}[X_1]$ (resp. $α>|\mathbb{E}[X_1]|$) the set $\mathcal{H}_α$ (resp. $\mathcal{Z}_α$) is non-empty, closed, unbounded above and below, stationary, and regenerative. The collections $(\mathcal{H}_α)_{α> \mathbb{E}[X_1]}$ and $(\mathcal{Z}_α)_{α> |\mathbb{E}[X_1]|}$ are increasing in $α$ 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 $(\sup\{t < 0 : t \in \mathcal{H}_α\})_{α> \mathbb{E}[X_1]}$ 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 $(\sup\{t < 0 : t \in \mathcal{Z}_α\})_{α> |β|}$ when $(X_t)_{t \in \mathbb{R}}$ is a (two-sided) Brownian motion with drift $β$.
