The Gerber-Shiu discounted penalty function in the stationary renewal risk model.

*(English)*Zbl 1072.91027The 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.

Reviewer: Silvia Curteanu (Iaşi)

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

60K05 | Renewal theory |

91B28 | Finance etc. (MSC2000) |

##### Keywords:

Sparre Andersen model; discounted penalty function; stationary renewal risk process; ordinary renewal risk model; defective renewal equation; surplus prior to ruin; deficit at ruin; Laplace transform; Lundberg’s fundamental equation
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\textit{G. E. Willmot} and \textit{D. C. M. Dickson}, Insur. Math. Econ. 32, No. 3, 403--411 (2003; Zbl 1072.91027)

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