Projects per year

### 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 language | English |
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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 | 97-186 |

Number of pages | 90 |

ISBN (Electronic) | 9783642314070 |

ISBN (Print) | 9783642314063 |

DOIs | |

Publication status | Published - 2013 |

### Publication series

Name | Lecture Notes in Mathematics |
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Publisher | Springer |

Volume | 2061 |

ISSN (Print) | 0075-8434 |

## Fingerprint 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 Self-Similar Markov and Levy

1/04/08 → 30/06/08

Project: Research council

### ANALYTICAL PROPERTIES OF SCALE FUNCTIONS

Engineering and Physical Sciences Research Council

10/12/07 → 9/12/08

Project: Research council

## 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). Springer. https://doi.org/10.1007/978-3-642-31407-0_2