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Probab. Theory Relat. Fields (2011) 150:691708 DOI 10.1007/s00440-010-0289-4
Smoothness of scale functions for spectrally negative Lvy processes
T. Chan A. E. Kyprianou M. Savov
Received: 30 March 2009 / Revised: 2 March 2010 / Published online: 14 April 2010 Springer-Verlag 2010
Abstract Scale functions play a central role in the uctuation theory of spectrally negative Lvy processes and often appear in the context of martingale relations. These relations are often require excursion theory rather than It calculus. The reason for the latter is that standard It calculus is only applicable to functions with a sufcient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lvy measure. We place particular emphasis on spectrally negative Lvy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
Mathematics Subject Classication (2000) 60G51 91B30 60J45
T. ChanSchool of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK e-mail: [email protected]
A. E. Kyprianou (B)
Department of Mathematical Sciences,The University of Bath, Claverton Down, Bath BA2 7AY, UK e-mail: [email protected]
M. SavovDepartment of Statistics,Oxford University, 1 South Parks Road, Oxford OX1 3TG, UK e-mail: [email protected]
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1 Spectrally negative Lvy processes and scale functions
Suppose that X = {Xt : t 0} is a spectrally negative Lvy process with probabil
ities {Px : x
R
}. For convenience we shall write P in place of P0. That is to say a real valued stochastic process whose paths are almost surely right continuous with left limits and whose increments are stationary and independent. Let {Ft : t 0}
be the natural ltration satisfying the usual assumptions and denote by its Laplace exponent so that
E(e Xt ) = et()
where E denotes expectation with respect to P.It is well known that is nite for all 0, is...