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On q-scale functions of spectrally negative Lévy processes

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  02 September 2022

Anita Behme ,
David Oechsler  and
René Schilling
Anita Behme*
Affiliation:
Technische Universität Dresden
David Oechsler*
Affiliation:
Technische Universität Dresden
René Schilling*
Affiliation:
Technische Universität Dresden
*
*Postal address: Technische Universität Dresden, Institut für Mathematische Stochastik, Helmholtzstraße 10, 01069 Dresden, Germany.
*Postal address: Technische Universität Dresden, Institut für Mathematische Stochastik, Helmholtzstraße 10, 01069 Dresden, Germany.
*Postal address: Technische Universität Dresden, Institut für Mathematische Stochastik, Helmholtzstraße 10, 01069 Dresden, Germany.
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Abstract

We obtain series expansions of the q-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit q-scale functions. Moreover, we study smoothness properties of the q-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. (Prob. Theory Relat. Fields 150, 2011) for spectrally negative Lévy processes with Gaussian component or bounded variation.

Keywords

Laplace transformq-scale functionsseries expansionssmoothnessspectrally one-sided Lévy processesVolterra equationsBernstein functionsDoney’s conjecture

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60J45: Probabilistic potential theory

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 56 - 84
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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