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Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process

  • Yuri KabanovEmail author
  • Serguei Pergamenshchikov
Article
  • 51 Downloads

Abstract

We study the asymptotics of the ruin probability for a process which is the solution of a linear SDE defined by a pair of independent Lévy processes. Our main interest is a model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let \(\beta >0\) be the root of the cumulant-generating function \(H\) of the increment \(V_{1}\) of the log-price process. We show that the ruin probability admits the exact asymptotic \(Cu^{-\beta }\) as the initial capital \(u\to \infty \), assuming only that the law of \(V_{T}\) is non-arithmetic without any further assumptions on the price process.

Keywords

Ruin probabilities Dual models Price process Renewal theory Distributional equation Autoregression with random coefficients Lévy process 

Mathematics Subject Classification (2010)

60G51 

JEL Classification

G22 G23 

Notes

Acknowledgements

This research is funded by grant \(n^{\circ }\) 14.A12.31.0007 of the Government of the Russian Federation. The second author is partially supported by the Russian Federal Professor program (project \(n^{\circ }\) 1.472.2016/1.4) and the research project \(n^{\circ }\) 2.3208.2017/4.6 of the Ministry of Education and Science of the Russian Federation.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de Franche-ComtéBesançon, cedexFrance
  2. 2.Lomonosov Moscow State UniversityMoscowRussia
  3. 3.Institute of Informatics ProblemsFederal Research Center “Computer Science and Control” of Russian Academy of SciencesMoscowRussia
  4. 4.Laboratoire de Mathématiques Raphaël SalemUniversité de RouenRouenFrance
  5. 5.National Research Tomsk State UniversityTomskRussia

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