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On the Time Spent in the Red by a Refracted Lévy Risk Process

Part of: Stochastic processes Mathematical economics

Published online by Cambridge University Press:  30 January 2018

Jean-François Renaud
Jean-François Renaud*
Affiliation:
UQAM
*
∗ Postal address: Département de Mathématiques, Université du Québec à Montréal (UQAM), 201 Av. Président-Kennedy, Montréal, Québec, H2X 3Y7, Canada. Email address: [email protected]
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Abstract

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In this paper we introduce an insurance ruin model with an adaptive premium rate, henceforth referred to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the wealth process recovers. The analysis is focused mainly on the time a refracted Lévy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from [11] and [16], we identify the distribution of various functionals related to occupation times of refracted spectrally negative Lévy processes. For example, these results are used to compute both the probability of bankruptcy and the probability of Parisian ruin in this model with restructuring.

Keywords

Spectrally negative Lévy processrefractionoccupation timebankruptcyParisian ruin

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 91B30: Risk theory, insurance
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1171 - 1188
Copyright
© Applied Probability Trust 

References

Albrecher, H. and Lautscham, V. (2013). From ruin to bankruptcy for compound Poisson surplus processes. Astin Bull. 43, 213–243.Google Scholar
Albrecher, H., Cheung, E. C. K. and Thonhauser, S. (2011). Randomized observation periods for the compound Poisson risk model dividends. Astin Bull. 41, 645–672.Google Scholar
Albrecher, H., Cheung, E. C. K. and Thonhauser, S. (2013). Randomized observation periods for the compound Poisson risk model: the discounted penalty function. Scand. Actuarial J. 2013, 424–452.Google Scholar
Albrecher, H., Gerber, H. U. and Shiu, E. S. W. (2011). The optimal dividend barrier in the gamma-omega model. Europ. Actuarial J. 1, 43–55.Google Scholar
Czarna, I. and Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk process. J. Appl. Prob. 48, 984–1002.CrossRefGoogle Scholar
Egami, M. and Yamazaki, K. (2014). Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 1–22.CrossRefGoogle Scholar
Gerber, H. U., Shiu, E. S. W. and Yang, H. (2012). The Omega model: from bankruptcy to occupation times in the red. Europ. Actuarial J. 2, 259–272.Google Scholar
Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061), Springer, Heidelberg, pp. 97–186.CrossRefGoogle Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Kyprianou, A. E. (2013). Gerber–Shiu Risk Theory. Springer, Berlin.Google Scholar
Kyprianou, A. E. and Loeffen, R. L. (2010). Refracted Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 46, 24–44.Google Scholar
Kyprianou, A. E., Loeffen, R. and Pérez, J.-L. (2012). Optimal control with absolutely continuous strategies for spectrally negative Lévy processes, J. Appl. Prob. 49, 150–166.Google Scholar
Kyprianou, A. E., Pardo, J. C. and Pérez, J.-L. (2014). Occupation times of refracted Lévy processes. J. Theoret. Prob. 27, 1292–1315.CrossRefGoogle Scholar
Landriault, D., Renaud, J.-F. and Zhou, X. (2014). An insurance risk model with Parisian implementation delays. Method. Comput. Appl. Prob. 16, 583–607.CrossRefGoogle Scholar
Loeffen, R., Czarna, I. and Palmowski, Z. (2013). Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19, 599–609.CrossRefGoogle Scholar
Loeffen, R., Renaud, J.-F. and Zhou, X. (2014). Occupation times of intervals until first passage times for spectrally negative Lévy processes. Stoch. Process. Appl. 124, 1408–1435.Google Scholar