Adv. Appl. Prob. 37, 836–856 (2005) Printed in Northern Ireland  Applied Probability Trust 2005 ON A GENERAL CLASS OF RENEWAL RISK PROCESS: ANALYSIS OF THE GERBER–SHIU FUNCTION SHUANMING LI, ∗ University of Melbourne JOSÉ GARRIDO, ∗∗ Concordia University Abstract We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a K n distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities. Keywords: Sparre Andersen risk process; K n family of distributions; martingale; Laplace transform; generalized Lundberg equation; defective renewal equation 2000 Mathematics Subject Classification: Primary 62P05 Secondary 60K05 1. Introduction Much of the literature on ruin theory is concentrated on the classical risk model, in which claims occur as a Poisson process. Andersen (1957) let claims occur according to a more general renewal process and derived an integral equation for the corresponding ruin probability. Since then, random walks and queueing theory have provided a more general framework, which has led to explicit results in the case where interclaim times or claim severities have distributions related to the Erlang and phase-type, or more general, K n distributions, whose Laplace–Stieltjes transform is the ratio of a polynomial of degree k<n to a polynomial of degree n (see Willmot (1999)). Li and Garrido (2004) considered a risk process with interclaim times being independent and identically Erlang(n) distributed, for n ∈ N + ={1, 2,... }. It extends the classical risk model and the Erlang(2) model of Dickson (1998b), Dickson and Hipp (1998), (2001), and Cheng and Tang (2003). Gerber and Shiu (2003), (2005) further extended the theory to generalized Erlang(n) interarrival times (i.e. the distribution is the convolution of n exponential distributions, with possibly different parameters). The evaluation of the Gerber–Shiu expected discounted penalty function, first introduced in Gerber and Shiu (1998), is now one of the main research problems in ruin theory. Cheng and Tang (2003), Dickson (1998b), Dickson and Hipp (1998), (2001), Gerber and Shiu (2003), (2005), Li (2003), Li and Garrido (2004), and Lin (2003) first derived high-order integrodifferen- tial equations for the expected discounted penalty functions. When integrated iteratively, these produce defective renewal equations. Many ruin-related quantities, e.g. explicit asymptotic, Received 10 May 2004; revision received 21 March 2005. ∗ Postal address: Centre for Actuarial Studies, University of Melbourne, Victoria 3010, Australia. ∗∗ Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Quebec H4B 1R6, Canada. Email address: garrido@mathstat.concordia.ca 836 at https://www.cambridge.org/core/terms. https://doi.org/10.1239/aap/1127483750 Downloaded from https://www.cambridge.org/core. IP address: 34.228.24.229, on 25 May 2020 at 19:20:47, subject to the Cambridge Core terms of use, available