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Exact and asymptotic n-tuple laws at first and last passage


Reference:

Kyprianou, A. E., Pardo, J. C. and Rivero, V., 2010. Exact and asymptotic n-tuple laws at first and last passage. Annals of Applied Probability, 20 (2), pp. 522-564.

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    Official URL:

    http://dx.doi.org/10.1214/09-aap626

    Abstract

    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.

    Details

    Item Type Articles
    CreatorsKyprianou, A. E., Pardo, J. C. and Rivero, V.
    DOI10.1214/09-aap626
    Uncontrolled Keywordslast passage time, undershoot, conditioned levy process, n-tuple laws, levy process, fluctuation theory, overshoot, first passage time
    DepartmentsFaculty of Science > Mathematical Sciences
    RefereedYes
    StatusPublished
    ID Code21551

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