The Time at which a Lévy Process Creeps

Philip S Griffin (Syracuse University)
Ross A Maller (Australian National University)

Abstract


We show that if a Levy process creeps then the renewal function of the bivariate ascending ladder process satisfies certain continuity and differentiability properties. Then a left derivative of the renewal function is shown to be proportional to the distribution function of the time at which the process creeps over a given level, where the constant of proportionality is the reciprocal of the (positive) drift of the ascending ladder height process. This allows us to add the term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As an application, we derive a Laplace transform identity which generalises the second factorization identity. We also relate Doney and Kyprianou's extension of Vigon's equation amicale inversee to creeping. Some results concerning the ladder process, including the second factorization identity, continue to hold for a general bivariate subordinator, and are given in this generality.

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Pages: 2182-2202

Publication Date: November 13, 2011

DOI: 10.1214/EJP.v16-945

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