### Abstract

Original language | English |
---|---|

Pages (from-to) | 691-708 |

Number of pages | 18 |

Journal | Probability Theory and Related Fields |

Volume | 150 |

Issue number | 3-4 |

Early online date | 13 Apr 2010 |

DOIs | |

Publication status | Published - 1 Aug 2011 |

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### Cite this

*Probability Theory and Related Fields*,

*150*(3-4), 691-708. https://doi.org/10.1007/s00440-010-0289-4

**Smoothness of scale functions for spectrally negative Lévy processes.** / Chan, T; Kyprianou, Andreas E; Savov, M.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 150, no. 3-4, pp. 691-708. https://doi.org/10.1007/s00440-010-0289-4

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TY - JOUR

T1 - Smoothness of scale functions for spectrally negative Lévy processes

AU - Chan, T

AU - Kyprianou, Andreas E

AU - Savov, M

PY - 2011/8/1

Y1 - 2011/8/1

N2 - Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often require excursion theory rather than Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient 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 Lévy measure. We place particular emphasis on spectrally negative Lévy 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.

AB - Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often require excursion theory rather than Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient 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 Lévy measure. We place particular emphasis on spectrally negative Lévy 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.

UR - http://www.scopus.com/inward/record.url?scp=80051666798&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/s00440-010-0289-4

U2 - 10.1007/s00440-010-0289-4

DO - 10.1007/s00440-010-0289-4

M3 - Article

VL - 150

SP - 691

EP - 708

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -