The theory of scale functions for spectrally negative lévy processes

A. Kuznetsov, A.E. Kyprianou, V. Rivero

Research output: Chapter in Book/Report/Conference proceedingOther chapter contribution

82 Citations (Scopus)
85 Downloads (Pure)

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).
Original languageEnglish
Title of host publicationLévy Matters II
Subtitle of host publicationRecent Progress in Theory and Applications: Fractional Lévy Fields, and Scale Functions
Place of PublicationBerlin
PublisherSpringer
Pages97-186
Number of pages90
ISBN (Electronic)9783642314070
ISBN (Print)9783642314063
DOIs
Publication statusPublished - 2013

Publication series

NameLecture Notes in Mathematics
PublisherSpringer
Volume2061
ISSN (Print)0075-8434

Fingerprint

Scale Function
Lévy Process
Excursion Theory
Fluctuations (theory)
Infinitely Divisible Distribution
Process Calculi
Stochastic Calculus
Semimartingale
Calculus
Fluctuations
Decompose
Necessary

Cite this

Kuznetsov, A., Kyprianou, A. E., & Rivero, V. (2013). The theory of scale functions for spectrally negative lévy processes. In Lévy Matters II: Recent Progress in Theory and Applications: Fractional Lévy Fields, and Scale Functions (pp. 97-186). (Lecture Notes in Mathematics; Vol. 2061). Berlin: Springer. https://doi.org/10.1007/978-3-642-31407-0_2

The theory of scale functions for spectrally negative lévy processes. / Kuznetsov, A.; Kyprianou, A.E.; Rivero, V.

Lévy Matters II: Recent Progress in Theory and Applications: Fractional Lévy Fields, and Scale Functions. Berlin : Springer, 2013. p. 97-186 (Lecture Notes in Mathematics; Vol. 2061).

Research output: Chapter in Book/Report/Conference proceedingOther chapter contribution

Kuznetsov, A, Kyprianou, AE & Rivero, V 2013, The theory of scale functions for spectrally negative lévy processes. in Lévy Matters II: Recent Progress in Theory and Applications: Fractional Lévy Fields, and Scale Functions. Lecture Notes in Mathematics, vol. 2061, Springer, Berlin, pp. 97-186. https://doi.org/10.1007/978-3-642-31407-0_2
Kuznetsov A, Kyprianou AE, Rivero V. The theory of scale functions for spectrally negative lévy processes. In Lévy Matters II: Recent Progress in Theory and Applications: Fractional Lévy Fields, and Scale Functions. Berlin: Springer. 2013. p. 97-186. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-642-31407-0_2
Kuznetsov, A. ; Kyprianou, A.E. ; Rivero, V. / The theory of scale functions for spectrally negative lévy processes. Lévy Matters II: Recent Progress in Theory and Applications: Fractional Lévy Fields, and Scale Functions. Berlin : Springer, 2013. pp. 97-186 (Lecture Notes in Mathematics).
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