Interference In The Renewal Risk Model, The Dividend

For the classical risk model and a constant dividend barrier, Lin et al. [1] have studied the discounted penalty function at ruin, which is an important tool to quantify the riskiness of the barrier strategy. The analysis of the discounted penalty function in Lin et al. [1] was recently generalized to a Sparre Andersen risk model with generalized Erlang(n)-distributed inter-claim times by Li and Garrido [2]. A second important quantity in assessing the quality of a dividend barrier strategy is the distribution of the discounted sum of dividend payments until ruin.The purpose of this thesis is to present some results on the distribution of dividend payments until ruin and the Gerber-Shiu function under a Sparre Andersen risk process perturbed by diffusion with generalized Erlang(n)-distributed inter-claim times. We derive integro-differential equations for the Gerber-Shiu function, and the moment-generation function and the mth moment of the sum of the discounted dividend payments until ruin.As we all know, the boundary conditions are important here. In Chapter 1, we consider the perturbed Sparre Andersen risk process under a threshold dividend strategy. We adopt a similar approach to those used in Albrecher et al. [4], i.e., decomposing every inter-occurrence time with generalized Erlang(n) distribution into the independent sum of n exponential random variables with parametersλ₁,λ₂,...λ_n, then, with the results in Wan [3] we get the boundary conditions.In Chapter 2, we consider the perturbed Sparre Andersen risk process which is compounded by a geometric Brownian motion under a constant dividend strategy. As we all know, The standard Brownian motion can be seen as a limitation of a compound Poisson process, the proof of which can be found in Li [5]. Using the same arguments as in Li [5], we derive the boundary conditions on the dividend line.