# zbMATH — the first resource for mathematics

The Gerber-Shiu discounted penalty function in the stationary renewal risk model. (English) Zbl 1072.91027
The aim of this article is to investigate various properties associated with the stationary renewal risk process. In the introductory Section 1, the authors review the ordinary renewal risk model, the stationary (equilibrium) renewal risk process, the invariance property between the stationary renewal risk and the classical models, the discounted penalty function (hereafter DPF) introduced by H. Gerber and E. Shiu (1998) for the stationary renewal risk model, and the defective renewal equation satisfied by DPF. The paper also expresses the same DPF for the ordinary renewal risk model and shows in Section 2 that Gerber and Shiu’s DPF in the stationary renewal risk model can be expressed in terms of the same DPF for the ordinary renewal risk model; this relationship is proved to unify and generalizes known special cases. Section 3 finds useful and simplified connections between the deficit at ruin and surplus prior to ruin. The final Section 4 recovers the original Gerber-Shiu’s relationship between stationary and ordinary renewal risk models, and the invariance property between the stationary renewal risk model and the classical Poisson model with respect to the ruin probability is generalized substantially.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60K05 Renewal theory 91B28 Finance etc. (MSC2000)
Full Text:
##### References:
 [1] Asmussen, S., 2000. Ruin Probabilities. World Scientific, Singapore. · Zbl 0960.60003 [2] Cheng, Y.; Tang, Q., Moments of the surplus before ruin and the deficit at ruin in the Erlang (2) risk process, North American actuarial journal, 7, 1-12, (2003) · Zbl 1084.60544 [3] Dickson, D.; Hipp, C., On the time to ruin for Erlang (2) risk processes, Insurance: mathematics and economics, 29, 333-344, (2001) · Zbl 1074.91549 [4] Drekic, S., Dickson, D., Stanford, D., Willmot, G., 2003. On the distribution of the deficit at ruin when claims are phase-type. Scandinavian Actuarial Journal, in press. · Zbl 1142.62088 [5] Dufresne, F.; Gerber, H., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance: mathematics and economics, 7, 193-199, (1988) · Zbl 0674.62072 [6] Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events. Springer, Berlin. [7] Gerber, H.; Shiu, E., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: mathematics and economics, 21, 129-137, (1997) · Zbl 0894.90047 [8] Gerber, H.; Shiu, E., On the time value of ruin, North American actuarial journal, 2, 48-78, (1998) · Zbl 1081.60550 [9] Grandell, J., 1991. Aspects of Risk Theory. Springer, New York. · Zbl 0717.62100 [10] Karlin, S., Taylor, H., 1975. A First Course in Stochastic Processes, 2nd ed. Academic Press, New York. · Zbl 0315.60016 [11] Lin, X.; Willmot, G., Analysis of a defective renewal equation arising in ruin theory, Insurance: mathematics and economics, 25, 63-84, (1999) · Zbl 1028.91556 [12] Lin, X.; Willmot, G., The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: mathematics and economics, 27, 19-44, (2000) · Zbl 0971.91031 [13] Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., 1999. Stochastic Processes for Insurance and Finance. Wiley, Chichester, UK. · Zbl 0940.60005 [14] Willmot, G., Lin, X., 2001. Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, New York. · Zbl 0962.62099 [15] Willmot, G., Dickson, D., Drekic, S., Stanford, D., 2002. The deficit at ruin in the stationary renewal risk model. Institute of Insurance and Pension Research Report 02-10, University of Waterloo, Waterloo, Ont., Canada.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.