Threshold Dividend Risk Model

Risk theory is a focus of actuarial mathematics, it has developed promptly in recent ten years. Ruin theory is the keystone of risk theory. In this paper, the classical compound Poisson risk model and Erlang(n) risk model are considered in essence, the dividend barrier strategies are discussed based on the above two models. The most novel dividend strategy, which is the so-called threshold dividend strategy, is investigated.1. Based on the classical compound Poisson risk model, we extend the constant threshold barrier to the linear dividend barrier. The Gerber-Shiu discounted penalty function and its integro-differential equations are studied. The solutions of the equations are derived by Laplace transform. And also, we discuss the Lundberg's equation under the linear dividend barrier. Lastly, the partial integro-differential equations for the discounted penalty function are given.2. Under the constant threshold dividend barrier, we generalize the classical risk model to Erlang(n) risk processes. Two integro-differential equations of Gerber-Shiu discounted penalty function are gained by utilizing the strong Markov property of the Erlang(n) process at claim instants. By the theory of differential equation and stochastic processes, we get the solutions for two integro-differential equations. Then the relations between two solutions are discussed. In the end, two quantities related to ruin in Erlang(2) risk process with threshold dividend barrier are obtained, and the numerical results are calculated respectively when the claim sizes distribution are exponential and mixed exponential. Moreover, we have used the numerical figures to describe the changes of the ruin quantities.3. For the stochastic factor of market, we add an independent diffusion or Wienerprocess {W(t),t≥0} to the classical compound Poisson risk model, thus the classical risk model is extended to diffusion risk model. Then we study Gerber-Shiu...