Garrido, José and Morales, Manuel (2005) *On the Expected Discounted Penalty Function for Lévy Risk Processes.* Technical Report. Concordia University. Department of Mathematics & Statistics, Montreal, Quebec.

| PDF - Published Version 255Kb |

## Abstract

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, they studied the joint distribution of the time of ruin, the surplus before ruin and the deficit at ruin [Gerber and Shiu (1997, 1998a, 1998b), Gerber and Landry (1998)]. They work with the classical and the perturbed risk models and hint that their results can be extended to gamma and inverse Gaussian risk processes.

In this paper we work out this extension in the context of a more general risk model. The construction of Dufresne et al. (1991) is based on a non–negative, non–increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. Our aim is to extend their work to a generalized risk model driven by an increasing L´evy process. This first paper presents the results for the case when the aggregate claims process is a subordinator. Embedded in this wide family of risk models we find the gamma, inverse Gaussian and generalized inverse Gaussian

processes.

Divisions: | Concordia University > Faculty of Arts and Science > Mathematics and Statistics |
---|---|

Item Type: | Monograph (Technical Report) |

Authors: | Garrido, José and Morales, Manuel |

Series Name: | Department of Mathematics & Statistics. Technical Report No. 6/05 |

Corporate Authors: | Concordia University. Department of Mathematics & Statistics |

Institution: | Concordia University |

Date: | November 2005 |

ID Code: | 6672 |

Deposited By: | DIANE MICHAUD |

Deposited On: | 03 Jun 2010 15:56 |

Last Modified: | 08 Dec 2010 18:22 |

References: | Abramowitz, M. and Stegun, I. (1970). Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables.
Asmussen, S. (2000). Ruin Probabilities. Advanced Series on Statistical Science and Applied Probability. World Scientific. Barndorff-Nielsen, O.E. and Halgreen, C. (1977). Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete. 38. pp. 439–455. Barndorff-Nielsen, O.E., Mikosh, T. and Resnick, S., ed. (2001). Lévy Processes-Theory and Applications. Birk¨auser. Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics.121. Cambridge University Press. Bertoin, J. and Doney, R.A. (1994). Cram´er’s estimate for L´evy processes. Statistics and Probability Letters. (21). pp. 363–365. Bowers, M., Gerber, H., Hickman, J., Jones, D. and Nesbitt, C. (1997)Actuarial Mathematics, 2nd edition, Society of Actuaries, Schaumburg. Chaubey, Y., Garrido, J. and Trudeau, S. (1998). On the computation of aggregate claims distributions: some new approximations. Insurance: Mathematics and Economics. 23. pp. 215–230. Chiu, S.N. and Yin, C. (2005) “Passage times for a spectrally negative Lévy process with applications to risk theory”. Bernoulli . 11(3). pp. 511- 522. Drekic, S.; Stafford, J. E. and Willmot, G. E. (2004). Symbolic Calculation of the Moments of the Time of Ruin. Insurance: Mathematics and Economics. (34). pp. 109-120. Doney, R. A. and Kyprianou, A. E. (2005). Overshoots and Undershoots of Lévy Processes. To appear in Annals of Applied Probability. Dufresne, F. and Gerber, H. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance:Mathematics and Economics. 10. pp. 51–59. Dufresne, F., Gerber, H.U. and Shiu, E.S.W. (1991). Risk theory with the gamma process. ASTIN Bulletin. 21(2). pp. 177–192. Furrer, H.J. (1998). Risk processes perturbed by a �–stable Lévy motion. Scandinavian Actuarial Journal. (1). pp. 59–74. Furrer, H.J., Michna, Z. and Weron, A. (1997). Stable Lévy motion approximation in collective risk theory. Insurance: Mathematics and Economics. 20. pp. 97–114. Gerber, H.U. and Landry, B. (1998). On a discounted penalty at ruin in a jump–diffusion and the perpetual put option. Insurance: Mathematics and Economics. (22). pp. 263–276. Gerber, H.U. and Shiu, E.S.W. (1997). The joint istribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics. 21. pp. 129–137. Gerber, H.U. and Shiu, E.S.W. (1998a). On the time value of ruin. North American Actuarial Journal. 2(1). pp. 48–78. Gerber, H.U. and Shiu, E.S.W. (1998b). Pricing perpetual options for jump processes. North American Actuarial Journal. 2(3). pp. 101–112. Grandell, J. (1991). Aspects of Risk Theory. Springer Series in Statistics. Springer–Verlag. Huzak, M.; Perman, M.; Sikic, H. and Vondracek, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. The Annals of Applied Probability. (14)3. pp. 1378–1397. Janicki, A. and Weron, A. (1994). Simulation and Chaotic Behavior of �–stable Stochastic Processes. Monographs and Textbooks in Pure and Applied Mathematics. 178. Dekker. Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics. 9. Springer–Verlag. Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001). Modern Actuarial Risk Theory. Kluwer Academic Publishers. Kl¨uppelberg, C.; Kyprianou, A.E. and Maller, R.A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. The Annals of Applied Probability. (14)4. pp. 1766–1801. Madan, D.B., Carr, P. and Chang, E.C. (1998). The variance gamma process and option pricing. European Finance Review. 2. pp. 79–105. Morales, M. (2004). Risk theory with the generalized inverse Gaussian L´evy process. ASTIN Bulletin. (2). pp. 361–377. Morales, M. and Schoutens, W. (2003). “A risk model driven by Lévy processes.” Applied Stochastic Models in Business and Industry. 19. pp. 147–167. Politis, K. and Pitts, S. (2005). Approximations for the Deficit at Ruin and the Mean Ruin Time in the Classical Poisson Model. Working Paper. University of Cambridge, England. Sato, K.I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. Yang, H. and Zhang, L. (2001). Spectrally negative Lévy processes with applications in risk theory. Advances in Applied Probability. 33(1). pp. 281– 291. Zoloratev, M.V. (1964). “The first passage time of a level and the behavior at infinity for a class of processes with independent increments. Probability Theory and its Applications. 9(4). pp. 653–662. |

Repository Staff Only: item control page