Models Involving Erlang Components
This chapter complements the material of Chap. 4 in the sense that explicit identification of requisite quantities is possible with stronger parametric assumptions about either the (marginal) distribution of the claim sizes or that of the interclaim times. Convenient parametric assumptions involve Erlang (or Coxian) distributions, due to the generality and mathematical tractability inherent in the use of such models. Sect. 5.1 directly generalizes the classical Poisson model of Chap. 3 by essentially assuming a (marginal) Erlang form for the interclaim times. A Lagrange polynomial approach together with Laplace transform arguments is used to explicitly identify all components of the Gerber-Shiu function. The independent model with an arbitrary interclaim time distribution and exponential claim sizes is then considered in Sect. 5.2. The conditioning argument of Sect. 5.2 is then extended in Sect. 5.3 to a model where the (marginal) claim size distribution is assumed to be of Coxian form. In particular, identification of the Gerber-Shiu function is shown to be possible for special choices of the penalty function, and more generally for an arbitrary penalty function of the deficit. Finally, the Coxian assumption of Sect. 5.3 is replaced by a (possibly infinite) mixture of Erlang distributions, resulting in tractable results for ruin probabilities, the deficit at ruin, and related quantities.