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Exit Problems for Spectrally Negative L evy Processes and Applications to Russian, American and Canadized

Summary: Exit Problems for Spectrally Negative Levy Processes
and Applications to Russian, American and Canadized
F.Avram  & A.E. Kyprianou y & M.R.Pistorius z
Universite de Pau Utrecht University Utrecht University
We consider spectrally negative Levy process and determine the joint Laplace trans-
form of the exit time and exit position from an interval containing the origin of the
process re ected in its supremum. In the literature of uid models, this stopping time
can be identi ed as the time to bu er-over ow. The Laplace transform is determined
in terms of the scale functions that appear in the two sided exit problem of the given
Levy process. The obtained results together with existing results on two sided exit
problems are applied to solving optimal stopping problems associated with the pricing
of American and Russian options and their Canadized versions.
AMS subject classi cation (2000). Primary 60J99; secondary 60G40, 91B70.
Key words and phrases. Re ected Levy processes, Exit problems, scale functions,
American option, Russian option, Canadized option, optimal stopping.
1 Introduction
In this paper we consider the class of spectrally negative Levy 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


Source: Avram, Florin - Laboratoire de Mathématiques Appliquées, Université de Pau et des Pays de l'Adour


Collections: Mathematics