Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T19:58:12.564Z Has data issue: false hasContentIssue false
  • English
  • Français

The Joint Density of the Surplus Before and After Ruin in the Sparre Andersen Model

Part of: Special processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Susan M. Pitts  and
Konstadinos Politis
Susan M. Pitts*
Affiliation:
University of Cambridge
Konstadinos Politis*
Affiliation:
University of Piraeus
*
∗Postal address: Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK. Email address: s.pitts@statslab.cam.ac.uk
∗∗Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou St., Piraeus 18534, Greece.
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.

Gerber and Shiu (1997) have studied the joint density of the time of ruin, the surplus immediately before ruin, and the deficit at ruin in the classical model of collective risk theory. More recently, their results have been generalised for risk models where the interarrival density for claims is nonexponential, but belongs to the Erlang family. Here we obtain generalisations of the Gerber-Shiu (1997) results that are valid in a general Sparre Andersen model, i.e. for any interclaim density. In particular, we obtain a generalisation of the key formula in that paper. Our results are made more concrete for the case where the distribution between claim arrivals is phase-type or the integrated tail distribution associated with the claim size distribution belongs to the class of subexponential distributions. Furthermore, we obtain conditions for finiteness of the joint moments of the surplus before ruin and the deficit at ruin in the Sparre Andersen model.

Keywords

Sparre Andersen modelsurplus prior to ruindeficit at ruinladder heightWiener-Hopf factorsphase-type distributionsubexponential distribution

MSC classification

Secondary: 60K10: Applications (reliability, demand theory, etc.) 91B30: Risk theory, insurance 60K05: Renewal theory
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 695 - 712
Copyright
Copyright © Applied Probability Trust 2007 

References

Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
Asmussen, S. (1992). Phase-type representations in random walk and queueing problems. Ann. Prob. 20, 772–789.CrossRefGoogle Scholar
Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.Google Scholar
Asmussen, S. and Klüppelberg, C. (1996). Large deviation results for subexponential tails, with applications to insurance risk. Stoch. Process. Appl. 64, 105–125.Google Scholar
Badescu, A. et al. (2005a). Risk processes analysed as fluid queues. Scand. Actuarial J. 2005, 127–141.Google Scholar
Badescu, A. et al. (2005b). The surplus prior to ruin and the deficit at ruin for a correlated risk process. Scand. Actuarial J. 2005, 433–445.Google Scholar
Cai, J. (2004). Ruin probabilities and penalty functions with stochastic rates of interest. Stoch. Process. Appl. 112, 53–78.Google Scholar
Cheng, Y. and Tang, Q. (2003). Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. N. Amer. Actuarial J. 7, 1–12.Google Scholar
Cheng, Y. B., Tang, Q. H. and Yang, H. L. (2002). Approximations for moments of deficit at ruin with exponential and subexponential claims. Statist. Prob. Lett. 59, 367–378.Google Scholar
Dickson, D. C. M. (1992). On the distribution of the surplus prior to ruin. Insurance Math. Econom. 11, 191–207.Google Scholar
Dickson, D. C. M. (1993). On the distribution of the claim causing ruin. Insurance Math. Econom. 12, 143–154.Google Scholar
Dickson, D. C. M. and Drekic, S. (2004). The Joint distribution of the surplus prior to ruin and the deficit at ruin in some Sparre Andersen models. Insurance Math. Econom. 34, 97–107.Google Scholar
Dickson, D. C. M. and Hipp, C. (2000). Ruin problems for phase-type(2) risk processes. Scand. Actuarial J. 2000, 147–167.CrossRefGoogle Scholar
Dickson, D. C. M. and Hipp, C. (2001). On the time to ruin for Erlang(2) risk processes. Insurance Math. Econom. 29, 333–344.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
Dufresne, F. and Gerber, H. U. (1988). The surpluses immediately before and at ruin, and the amount of claim causing ruin. Insurance Math. Econom. 7, 193–199.Google Scholar
Embrechts, P. and Veraverbecke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosh, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Gerber, H. U. and Shiu, E. (1997). The Joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance Math. Econom. 21, 129–137.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value at ruin. N. Amer. Actuarial J. 2, 48–78.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (2003a). Discussion on: ‘Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process’. N. Amer. Actuarial J. 7, 117–119.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (2003b). Comment on: ‘Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process’. N. Amer. Actuarial J. 7(4), 96–101.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (2005). The time value of ruin in a Sparre Andersen model. N. Amer. Actuarial J. 9, 49–84.Google Scholar
Gerber, H. U., Goovaerts, M. J. and Kaas, R. (1987). On the probability and severity of ruin. Astin Bull. 17, 151–163.CrossRefGoogle Scholar
Gut, A. (1987). Stopped Random Walks: Limit Theorems and Applications. Springer, Berlin.Google Scholar
Kennedy, J. (1994). Understanding the Wiener–Hopf factorisation for the simple random walk. J. Appl. Prob. 31, 561–563.Google Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82, 259–269.Google Scholar
Li, S. M. and Garrido, J. (2004). On ruin for the Erlang(n) risk process. Insurance Math. Econom. 34, 391–408.CrossRefGoogle Scholar
Li, S. M. and Garrido, J. (2005). On a general class of renewal risk process: analysis of the Gerber–Shiu function. Adv. Appl. Prob. 37, 836–856.Google Scholar
Lin, X. S. and Willmot, G. E. (2000). The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance Math. Econom. 27, 19–44.Google Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. Dover, New York.Google Scholar
Picard, P. and Lefèvre, C. (1998). The moments of ruin time in the classical risk model with discrete claim size distribution. Insurance Math. Econom. 23, 157–172.Google Scholar
Picard, P. and Lefèvre, C. (2003). Probability of ruin in a discrete-time ruin model. J. Appl. Prob. 40, 543–556.CrossRefGoogle Scholar
Teugels, J. L. (1975). The class of subexponential distributions. Ann. Prob. 3, 1000–1011.Google Scholar
Willmot, G. E. (2002). Compound geometric residual lifetime distributions and the deficit at ruin. Insurance Math. Econom. 30, 421–438.Google Scholar