EXACT AND ASYMPTOTIC n-TUPLE LAWS AT FIRST AND LAST PASSAGE

• 成果类型: Article
• 署名作者: Kyprianou, A. E.; Pardo, J. C.; Rivero, V.
• 署名单位: University of Bath
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/09-AAP626
• 发表日期: 2010
• 页码: 522-564
• 关键词: Levy processes; overshoots; ruin

摘要

Understanding the space time features of how a Levy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes, to name but a few. In Doney and Kyprianou [Ann. Appl. Probab. 16 (2006) 91-106] a new quintuple law was established for a general Levy process at first passage below a fixed level. In this article we use the quintuple law to establish a family of related joint laws, which we call n-tuple laws, for Levy processes, Levy processes conditioned to stay positive and positive self-similar Markov processes at both first and last passage over a fixed level. Here the integer n typically ranges from three to seven. Moreover, we look at asymptotic overshoot and undershoot distributions and relate them to overshoot and undershoot distributions of positive self-similar Markov processes issued from the origin. Although the relation between the n-tuple laws for Levy processes and positive self-similar Markov processes are straightforward thanks to the Lamperti transformation, by interplaying the role of a (conditioned) stable processes as both a (conditioned) Levy processes and a positive self-similar Markov processes, we obtain a suite of completely explicit first and last passage identities for so-called Lamperti-stable Levy processes. This leads further to the introduction of a more general family of Levy processes which we call hypergeometric Levy processes, for which similar explicit identities may be considered.