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2014 | OriginalPaper | Chapter

10. Ruin Problems and Gerber–Shiu Theory

Author : Andreas E. Kyprianou

Published in: Fluctuations of Lévy Processes with Applications

Publisher: Springer Berlin Heidelberg

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Abstract

A natural generalisation of the classical Cramér–Lundberg insurance risk model is a spectrally negative Lévy process; also called a Lévy insurance risk process. In this chapter, we shall return to the first-passage problem for Lévy processes, which has already been studied in Chap. 7, and look at the role it plays in a family of problems which have proved to be an extensive topic of research in the actuarial literature. Many of the problems we shall consider are inspired by the longstanding collaborative contributions of Hans Gerber and Elias Shiu, thereby motivating the title of this chapter.

We shall start by reviewing classical results that have already been treated implicitly, if not explicitly, earlier in this book. Largely, this concerns the exact and asymptotic distributions of overshoots and undershoots of the Lévy insurance risk process at ruin. Thereafter, we shall turn our attention to more complex models of insurance risk in which dividends or tax are paid out of the insurance risk process, thereby adjusting its trajectory. In this setting, a number of identities concerning ruin of the resulting adjusted process, as well as the dividends or tax paid out until ruin, are investigated.

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Recall from the discussion following Lemma 8.2 that W ^(q) is continuously differentiable when X has paths of unbounded variation and otherwise it is continuously differentiable if and only if the Lévy measure of X has no atoms.

See for example the discussion on p. 80 of Gerber and Shiu (2006b) which also makes reference to “refraction” in the case of compound Poisson jumps. Gerber and Shiu (2006a) also use the terminology “refraction” for the case that X is a linear Brownian motion.

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Title: Ruin Problems and Gerber–Shiu Theory
Author: Andreas E. Kyprianou
Publisher: Springer Berlin Heidelberg
Book: Fluctuations of Lévy Processes with Applications
Print ISBN: 978-3-642-37631-3

Electronic ISBN: 978-3-642-37632-0

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-642-37632-0_10