FLUCTUATION THEORY AND EXIT SYSTEMS FOR POSITIVE SELF-SIMILAR MARKOV PROCESSES

• 成果类型: Article
• 署名作者: Chaumont, Loic; Kyprianou, Andreas; Carlos Pardo, Juan; Rivero, Victor
• 署名单位: Universite d'Angers; University of Bath; CIMAT - Centro de Investigacion en Matematicas
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP612
• 发表日期: 2012
• 页码: 245-279
• 关键词: recurrent extensions; entrance; LAWS

摘要

For a positive self-similar Markov process, X, we construct a local time for the random set, Theta, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set Theta and the process X sampled on the local time scale. The process (R, H) is described in terms of a ladder process linked to the Levy process associated to X via Lamperti's transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R, H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying Levy process oscillates.