Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-03T05:33:01.519Z Has data issue: false hasContentIssue false
  • English
  • Français

Ruin Probabilities for Two Classes of Risk Processes

Published online by Cambridge University Press:  17 April 2015

Shuanming Li  and
José Garrido
Shuanming Li
Affiliation:
Centre for Actuarial Studies, Faculty of Economics and Commerce, University of Melbourne, Parkville 3052, Victoria, Australia, Email: [email protected]
José Garrido
Affiliation:
Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montréal, Québec, H4B 1R7 Canada, Email: [email protected]
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 a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim inter-arrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the Kn family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.

Keywords

Compound Poisson processgeneralized Erlang distributionSparre Andersen risk modelintegro-differential equationsmartingalesgeneralized Lundberg fundamental equationsupremum distributionWiener processdiffusion perturbed risk model
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 1 , May 2005 , pp. 61 - 77
Copyright
Copyright © ASTIN Bulletin 2005

Footnotes

*

This research was funded by a 1SOA/CAS Ph.D. Grant and the 2Natural Sciences and Engineering Research Council of Canada (NSERC) operating grant OGP0036860.

References

Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
Asmussen, S. (1989) Risk theory in a Markovian environment. Scandinavian Actuarial Journal, 69–100.CrossRefGoogle Scholar
Cheng, Y. and Tang, Q. (2003) Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. North American Actuarial Journal, 7(1), 1–12.CrossRefGoogle Scholar
Cohen, J.W. (1982) The Single Server Queue, Rev. Ed., North-Holland, Amsterdam.Google Scholar
Dickson, D.C.M. (1998) On a class of renewal risk process. North American Actuarial Journal, 2(3), 60–68.CrossRefGoogle Scholar
Dickson, D.C.M. and Hipp, C. (1998) Ruin probabilities for Erlang(2) risk processes. Insurance: Mathematics and Economics, 22, 251–262.Google Scholar
Dickson, D.C.M. and Hipp, C. (2001) On the time to ruin for Erlang(2) risk processes. Insurance: Mathematics and Economics, 29, 333–344.Google Scholar
Gerber, H.U. and Landry, B. (1998) On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Mathematics and Economics, 22, 263–276.Google Scholar
Gerber, H.U. and Shiu, E.S.W. (1998) Pricing perpetual option for jump processes. North American Actuarial Journal, 2(3), 101–107.CrossRefGoogle Scholar
Gerber, H.U. and Shiu, E.S.W. (2003a) Discussion of Yebin Cheng and Qihe Tang’s “Moments of the surplus before ruin and the deficit at ruin”. North American Actuarial Journal, 7(3), 117–119.CrossRefGoogle Scholar
Gerber, H.U. and Shiu, E.S.W. (2003b) Discussion of Yebin Cheng and Qihe Tang’s “Moments of the surplus before ruin and the deficit at ruin”. North American Actuarial Journal, 7(4), 96–101.CrossRefGoogle Scholar
Gerber, H.U. and Shiu, E.S.W. (2005) On the time value of ruin in a Sparre Andersen risk process. North American Actuarial Journal, 9(2), in press.CrossRefGoogle Scholar
Li, S. (2003) Discussion of Yebin Cheng and Qihe Tang’s “Moments of the surplus before ruin and the deficit at ruin”. North American Actuarial Journal, 7(3), 119–122.CrossRefGoogle Scholar
Li, S. and Garrido, J. (2004) On ruin for the Erlang (n) risk process. Insurance: Mathematics and Economics, 34(3), 391–408.Google Scholar
Lin, X.S. (2003) Discussion of Yebin Cheng and Qihe Tang’s “Moments of the surplus before ruin and the deficit at ruin”. North American Actuarial Journal, 7(3), 122–124.Google Scholar
Willmot, G.E. (1999) A Laplace transform representation in a class of renewal queuing and risk process. Journal of Applied Probability, 36, 570–584.CrossRefGoogle Scholar
Yuen, K.C., Guo, J. and Wu, X. (2002) On a correlated aggregate claims model with Poisson and Erlang risk processes. Insurance: Mathematics and Economics, 31, 205–214.Google Scholar