×

The theory of scale functions for spectrally negative Lévy processes. (English) Zbl 1261.60047

Cohen, Serge (ed.) et al., Lévy matters II. Recent progress in theory and applications: fractional Lévy fields, and scale functions. Berlin: Springer (ISBN 978-3-642-31406-3/pbk; 978-3-642-31407-0/ebook). Lecture Notes in Mathematics 2061. Lévy Matters, 97-186 (2012).
“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.”
Spectrally negative Lévy processes are basically Lévy processes possessing only negative jumps (but excluding monotone sample paths). Hence, their Laplace transform is well-defined and can be used to define the scale function of the spectrally negative Lévy processes. The authors motivate extensively the introduction of scale functions which appear naturally, for instance, in optimal stopping or optimal control problems as well as ruin and queuing theory.
In the following, several properties of spectrally negative Lévy processes are recalled and the existence of scale functions is proved. The relation to excursion theory is addressed and a lot of properties of scale functions are derived such as fluctuation identities concerning first and last passage problems, asymptotic behaviour, concave-convex properties and smoothness. The corresponding proofs are partly sketched, focusing on the main ideas, and partly completely given. In the second part of the article, the authors study the analytic construction and the numeric analysis of scale functions. More precisely, scale functions can be constructed via the Wiener-Hopf factorization of the Lévy process, using special Bernstein functions or derived from another scale function and an associated parent process. However, in most cases an explicit expression for the inverse of the Laplace transform is not available and thus numerical methods have to be used to obtain the scale function of a given spectrally negative Lévy processes. For example, Filon’s method, the Gaver-Stehfest algorithm, Euler algorithm and the Talbot algorithm are described. Finally, numerical examples are presented.
For the entire collection see [Zbl 1252.60001].

MSC:

60G51 Processes with independent increments; Lévy processes
60G40 Stopping times; optimal stopping problems; gambling theory
60J45 Probabilistic potential theory
91B70 Stochastic models in economics
PDFBibTeX XMLCite
Full Text: DOI arXiv