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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 LévyKhintchine formula and its relationship to the LévyItô decomposition. We shall also touch on more general topics such as excursion theory and semimartingale 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 language  English 

Title of host publication  Lévy Matters II 
Subtitle of host publication  Recent Progress in Theory and Applications: Fractional Lévy Fields, and Scale Functions 
Place of Publication  Berlin 
Publisher  Springer 
Pages  97186 
Number of pages  90 
ISBN (Electronic)  9783642314070 
ISBN (Print)  9783642314063 
DOIs  
Publication status  Published  2013 
Publication series
Name  Lecture Notes in Mathematics 

Publisher  Springer 
Volume  2061 
ISSN (Print)  00758434 
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Dive into the research topics of 'The theory of scale functions for spectrally negative lévy processes'. Together they form a unique fingerprint.Projects
 2 Finished

Fluctuaction Theory of Positive SelfSimilar Markov and Levy
Kyprianou, A.
1/04/08 → 30/06/08
Project: Research council

ANALYTICAL PROPERTIES OF SCALE FUNCTIONS
Kyprianou, A.
Engineering and Physical Sciences Research Council
10/12/07 → 9/12/08
Project: Research council