Renewal risk processes

Abstract

In this chapter we consider a generalization of the classical ruin theoretic model. The generalization involves replacement of the Poisson process of claims by a more general renewal process. The model may be formulated as a particular random walk, and as such also allows for interpretation in terms of the equilibrium waiting time distribution in the G/G/l queue. These ideas are discussed in more detail in section 11.1, where various properties of the model are presented, including in particular Lundberg-type upper and lower bounds and a close relationship with the compound geometric distribution. Analytic complexities obviate the need for such bounds and approximations, and this approach has been followed by various authors such as Abate, Choudhury, and Whitt (1995). A more detailed analysis of this model may be found in Prabhu (1998), Cohen (1982), Resnick (1992), Ross (1996), or Feller (1971). See Grandell (1991, chapter 3), or Rolski et al (1999, section 6.5), for insurance applications.