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The Theory of Scale Functions for Spectrally Negative Lévy Processes

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Lévy Matters II

Part of the book series: Lecture Notes in Mathematics ((LEVY,volume 2061))

Abstract

The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative Lévy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Lévy processes, in particular a reasonable understanding of the Lévy–Khintchine formula and its relationship to the Lévy–Itô decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Lévy processes; (Bertoin, Lévy Processes (1996); Sato, Lévy Processes and Infinitely Divisible Distributions (1999); Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes and Their Applications (2006); Doney, Fluctuation Theory for Lévy Processes (2007)), Applebaum Lévy Processes and Stochastic Calculus (2009).

Mathematics Subject Classification 2000: Primary: 60G40, 60G51, 60J45 Secondary: 91B70

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Notes

  1. 1.

    Curious about the results in [27], Doney produced a number of specific examples of Lévy processes whose scale functions exhibited analytical behaviour that lead to his conjecture (personal communication with A.E. Kyprianou).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, Canada, M3J 1P3

    Alexey Kuznetsov

  2. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA1 2UU, UK

    Andreas E. Kyprianou

  3. Centro de Investigación en Matemáticas A.C., Calle Jalisco s/n, C.P. 36240, Guanajuato, Gto. Mexico

    Victor Rivero

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Kuznetsov, A., Kyprianou, A.E., Rivero, V. (2012). The Theory of Scale Functions for Spectrally Negative Lévy Processes. In: Lévy Matters II. Lecture Notes in Mathematics(), vol 2061. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31407-0_2

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