Bridges of Levy processes conditioned to stay positive

被引:11
|
作者
Uribe Bravo, Geronimo [1 ]
机构
[1] Univ Nacl Autonoma Mexico, Inst Matemat, Area Invest Cient, Mexico City 04510, DF, Mexico
关键词
Levy processes; Markovian bridges; Vervaat transformation; WEAK-CONVERGENCE; BROWNIAN BRIDGE; MEANDER; TIMES;
D O I
10.3150/12-BEJ481
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
We consider Kallenberg's hypothesis on the characteristic function of a Levy process and show that it allows the construction of weakly continuous bridges of the Levy process conditioned to stay positive. We therefore provide a notion of normalized excursions Levy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the Levy process conditioned to stay positive by using the weakly continuous bridges of the Levy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper semi-continuous versions. Using the bridges of the conditioned Levy process, the Durrett-Iglehart theorem stating that the Brownian bridge from 0 to 0 conditioned to remain above -epsilon converges weakly to the Brownian excursion as epsilon -> 0, is extended to Levy processes. We also extend the Denisov decomposition of Brownian motion to Levy processes and their bridges, as Well as Vervaat's classical result stating the equivalence in law of the Vervaat transform of a Brownian bridge and the normalized Brownian excursion.
引用
收藏
页码:190 / 206
页数:17
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