Abstract

this paper we consider the class of spectrally negative L'evy processes. These are real valued random processes with stationary independent increments which have no positive jumps. Amongst others Emery [11], Suprun [23], Bingham [4] and Bertoin [3] have all considered fluctuation theory for this class of processes. Such processes are often considered in the context of the theories of dams, queues, insurance risk and continuous branching processes; see for example [6, 4, 5, 19]. Following the exposition on two sided exit problems in Bertoin [3] we study first exit from an interval containing the origin for spectrally negative L'evy processes reflected in their supremum (equivalently spectrally positive L'evy processes reflected in their infimum). In particular we derive the joint Laplace transform of the time to first exit and the overshoot. The aforementioned stopping time can be identified in the literature of fluid models as the time to buffer overflow (see for example [1, 13]). Together Universit'e de Pau, e-mail: Florin.Avram@univ-pau.fr y Utrecht University, e-mail: kyprianou@math.uu.nl z Utrecht University, e-mail: pistorius@math.uu.nl 1 with existing results on exit problems we apply our results to certain optimal stopping problems that are now classically associated with mathematical finance. In sections 2 and 3 we introduce notation and discuss and develop existing results concerning exit problems of spectrally negative L'evy processes. In section 4 an expression is derived for the joint Laplace transform of the exit time and exit position of the reflected process from an interval containing the origin. This Laplace transform can be written in terms of scale functions that already appear in the solution to the two sided exit problem. In Section 5 we ou...

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