Hostname: page-component-7b7567c8c-2rsb6 Total loading time: 0 Render date: 2026-11-13T10:55:30.886Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations for risk processes with reinsurance

Part of:Limit theoremsMathematical economicsApplications

Published online by Cambridge University Press:  14 July 2016

Claudio Macci and
Gabriele Stabile
Claudio Macci*
Affiliation:
Università di Roma ‘Tor Vergata’
Gabriele Stabile*
Affiliation:
Università di Roma ‘La Sapienza’
*
∗ Postal address: Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica, I-00133 Rome, Italy. Email address: macci@mat.uniroma2.it
∗∗ Postal address: Dipartimento di Matematica per le Decisioni Economiche Finanziarie ed Assicurative, Università di Roma ‘La Sapienza’, Via del Castro Laurenziano 9, I-00161 Rome, Italy. Email address: gabriele.stabile@uniroma1.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We consider risk processes with reinsurance. A general family of reinsurance contracts is allowed, including proportional and excess-of-loss policies. Claim occurrence is regulated by a classical compound Poisson process or by a Markov-modulated compound Poisson process. We provide some large deviation results concerning these two risk processes in the small-claim case. Finally, we derive the so-called Lundberg estimate for the ruin probabilities and present a numerical example.

Keywords

Large deviationsrisk processreinsuranceruin probabilityLundberg estimate

MSC classification

Primary:60F10: Large deviations91B30: Risk theory, insurance62P05: Applications to actuarial sciences and financial mathematics

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006, pp. 713 - 728
Copyright
© Applied Probability Trust 2006 

References

Baldi, P. and Piccioni, M. (1999). A representation formula for the large deviation rate function for the empirical law of a continuous time Markov chain. Statist. Prob. Lett. 41, 107–115.Google Scholar
Borovkov, A. A. (1967). Boundary value problems for random walks and large deviations in function spaces. Theory Prob. Appl. 12, 575–595.Google Scholar
De Acosta, A. (1994). Large deviations for vector-valued Lévy processes. Stoch. Process. Appl. 51, 75–115.Google Scholar
Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, Boston, MA.Google Scholar
Duffield, N. G. and O'Connell, N. (1995). Large deviations and overflow probabilities for a single server queue, with applications. Math. Proc. Camb. Phil. Soc. 118, 363–374.Google Scholar
Hald, M. and Schmidli, H. (2004). On the maximisation of the adjustment coefficient under proportional reinsurance. ASTIN Bull. 34, 75–83.Google Scholar
Hipp, C. and Vogt, M. (2003). Optimal dynamic XL reinsurance. ASTIN Bull. 33, 193–207.CrossRefGoogle Scholar
Macci, C. and Torrisi, G. L. (2004). Asymptotic results for perturbed risk processes with delayed claims. Insurance Math. Econom. 34, 307–320.CrossRefGoogle Scholar
Macci, C., Stabile, G. and Torrisi, G. L. (2005). Lundberg parameters for non standard risk processes. Scand. Actuarial J. 2005, 417–432.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. L. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
Schmidli, H. (2001). Optimal proportional reinsurance policies in a dynamic setting. Scand. Actuarial J. 2001, 55–68.CrossRefGoogle Scholar
Schmidli, H. (2002). Asymptotics of ruin probabilities for risk processes under optimal reinsurance policies: the small claim case. Working paper 180, Laboratory of Actuarial Mathematics, University of Copenhagen.Google Scholar
Schmidli, H. (2004). Asymptotics of ruin probabilities for risk processes under optimal reinsurance and investment policies: the large claim case. Queueing Systems 46, 149–157.CrossRefGoogle Scholar
Waters, H. R. (1983). Some mathematical aspects of reinsurance. Insurance Math. Econom. 2, 17–26.Google Scholar