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Upper bounds on the expected time to ruin and on the expected recovery time

Part of: Special processes Mathematical economics

Published online by Cambridge University Press:  01 July 2016

Esther Frostig
Esther Frostig*
Affiliation:
University of Haifa
*
∗ Postal address: Department of Statistics, University of Haifa, Haifa 31905, Israel. Email address: frostig@stat.haifa.ac.il
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Abstract

It is shown that the time to ruin and the recovery time in a risk process have the same distribution as the busy period in a certain queueing system. Similarly, the deficit at the time of ruin is distributed as the idle period in a single-server queueing system. These duality results are exploited to derive upper bounds for the expected time to ruin and the expected recovery time as defined by Egídio dos Reis (2000). When the claim size is generally distributed, Lorden's inequality is applied to derive the bounds. When the claim-size distribution is of phase type, tighter upper bounds are derived.

Keywords

Risk processtime to ruinduration of negative surplusM/G/1 queueing systemG/M/1 queueing systemPH/PH/1 queueing systembusy periodidle periodLorden's inequalityphase-type distribution

MSC classification

Primary: 91B30: Risk theory, insurance
Secondary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 377 - 397
Copyright
Copyright © Applied Probability Trust 2004 

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References

Asmussen, S. (1989). Exponential families generated by phase-type distribution and other Markov life times. Scand. J. Statist. 16, 319–334.Google Scholar
Asmussen, S. (1992). Phase-type representations in random walk and queueing problems. Ann. Prob. 20, 772–789.Google Scholar
Asmussen, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2). World Scientific, River Edge, NJ.Google Scholar
Bäuerle, N., (1996). Some results about the expected ruin time in Markov-modulated risk models. Insurance Math. Econom. 18, 119–227.Google Scholar
Chang, J. T. (1994). Inequalities for the overshoot. Ann. Appl. Prob. 4, 1223–1233.Google Scholar
Dickson, D. C. M. and Egídio dos Reis, A. D. (1996). On the distribution of the duration of negative surplus. Scand. Actuarial J. 1996, 148–164.Google Scholar
Drekic, S., Dickson, D. C. M., Stanford, D. A. and Willmot, G. E. (2004). On the distribution of the deficit at ruin when claims are phase-type. Scand. Actuarial J. 2004, 105–120.Google Scholar
Egídio dos Reis, A. D. (1993). How long is the surplus below zero? Insurance Math. Econom. 12, 23–38.Google Scholar
Egídio dos Reis, A. D. (2000). On the moments of ruin and recovery times. Insurance Math. Econom. 27, 331–343.CrossRefGoogle Scholar
Esscher, F. (1932). On the probability function in the collective risk theory. Scand. Actuarial J. 15, 175–195.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1994). Option pricing by Esscher transforms. Trans. Soc. Actuaries 46, 99–140.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2, 48–72.Google Scholar
Lin, X. S. and Willmot, G. E. (2000). The moments of the time to ruin, the surplus before ruin, and the deficit at ruin. Insurance Math. Econom. 27, 19–44.Google Scholar
Lorden, G. (1970). On excess over the boundary. Ann. Math. Statist. 41, 520–527.CrossRefGoogle Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Prabhu, N. U. (1997). Foundations of Queueing Theory (Internat. Ser. Operat. Res. Manag. Sci. 7). Kluwer, Boston, MA.Google Scholar
Prabhu, N. U. (1998). Stochastic Storage Processes (Appl. Math. (New York) 15), 2nd edn. Springer, New York.Google Scholar
Ross, S. M. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Vyncke, D., Goovaerts, M. J., Kaas, R. and Dhaene, J. (2003). On the distribution of cash flows using Esscher transforms. J. Risk Insurance 70, 563–575.CrossRefGoogle Scholar
Willmot, G. E. and Lin, X. S. (2001). Lundberg Approximations for Compound Distributions with Insurance Applications (Lecture Notes Statist. 156). Springer, New York.Google Scholar