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2014 | OriginalPaper | Chapter

9. More on Scale Functions

Author : Andreas E. Kyprianou

Published in: Fluctuations of Lévy Processes with Applications

Publisher: Springer Berlin Heidelberg

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Abstract

In Chap. 8, we saw that it is possible to develop many fluctuation identities for spectrally negative Lévy processes in terms of scale functions. In this chapter, we continue in this vein and look in greater detail at the relationship between scale functions and potential measures of subordinators through the Wiener–Hopf factorisation. This will allow us to extract a number of additional analytical properties for scale functions as well as to offer a method for generating many examples of spectrally negative Lévy processes whose associated scale functions can be computed explicitly.

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previous chapter Exit Problems for Spectrally Negative Processes

next chapter Ruin Problems and Gerber–Shiu Theory

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We remind the reader that many examples can be found directly in Schilling et al. (2010) and, as inverse local times, in Borodin and Salminen (2002).

Borodin, A. and Salminen, P. (2002) Handbook of Brownian Motion—Facts and Formulae. 2nd Edition. Birkhäuser Basel. CrossRefMATH

Chazal, M., Kyprianou, A.E. and Patie, P. (2012) A transformation for Lévy processes with one-sided jumps and applications. To appear in Adv. Appl. Probab.

Hubalek, F. and Kyprianou, A.E. (2010) Old and new examples of scale functions for spectrally negative Lévy processes. In, Sixth Seminar on Stochastic Analysis, Random Fields and Applications, R. Dalang, M. Dozzi, and F. Russo (Eds.), Progress in Probability, 63. Birkhäuser, Basel, 119–146.

Konstantopoulos, T., Kyprianou, A.E. and Salminen, P. (2011) On the excursions of reflected local time processes and stochastic fluid queues. J. Appl. Probab. 48, 79–98. CrossRefMathSciNet

Kyprianou, A.E. and Rivero, V. (2008) Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13, 1672–1701. CrossRefMATHMathSciNet

Schilling, R., Song, R., and Vondraček, Z. (2010) Bernstein Functions. Theory and Applications. de Gruyter Studies in Mathematics, 37. Walter de Gruyter, Berlin. MATH

Title: More on Scale Functions
Author: Andreas E. Kyprianou
Publisher: Springer Berlin Heidelberg
Book: Fluctuations of Lévy Processes with Applications
Print ISBN: 978-3-642-37631-3

Electronic ISBN: 978-3-642-37632-0

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-642-37632-0_9