Periodic Dividends In A Dual Model With Perturbations

Since 1957. the dividend strategy was first introduced and discussed by De Finetti in the discrete time risk model. From then on, the issue of dividends has received remark-able attention in the insurance theory for actuaries. The issue of continuous dividend has gotten good research and development, while the issue of periodic dividend was first proposed in 2011 by Albrecher. In this paper, we discuss a dual model with diffusion. More precisely, we discuss two major dividend strategies in terms of periodic dividend: the barrier dividend strategy and the threshold dividend strategy. The main problems we discussed in this paper are the expected discounted dividend function and the Laplace transform of ruin time when the jump are exponential distribution and Erlang(n) distri-bution respectively.The first chapter is the introduction. This chapter is divided into two parts. The fist part introduces the development history of the insurance theory for actuaries, primarily about the surplus model with this paper.The second chapter is the model introduction. This chapter introduces the dual risk model with diffusion and the definition of the relevant dividend strategy. It also introduces the symbolic representation of two quantities in this paper:the expected discounted dividend strategy J(u; H) and the Laplace transform of ruin time M(u; H).The third, forth and fifth chapter are the main results of this paper. The main contents are as follows.First of all, we consider the model with periodic barrier dividend strategy when the jump is exponential distribution in the third chapter. According to the dividend level b, the Laplace transform of ruin time m(u;b) of the initial surplus u above and below the barrier b are divided into m1(u;b) and m2(u; b). Considering three possible cases in a very small time period [0, h], we have (1) there are dividends and no income. (2)there are incomes and no dividend. (3)there are no dividend and no income. Having incomes can also be divided into cases. One is that the surplus reaches above the barrier b while the other doesn't. Because of above discussion, a system of intergo-differential equations and the boundary conditions can be obtained. Moreover, we can obtain the analytical solution and exact expression of m(u; b).Then, we consider the model with periodic barrier dividend strategy when the jump is Erlang(n) distributed in the forth chapter. Similarly, the expected discounted dividend function V(u;b) is divided into VL(u; b) and Vu(u; b). We can derive the intergo-differential equation sets and the boundary conditions of V(u; b) by same methods. When solving the equation sets, we only need solve VL,1(u;b) and VU,1{u;b) because of V(u;b)= V1{u;b). The ordinary differential equation of V1(u;b) can be obtained by using the recursive thought. Moreover, we obtain the analytical solution of V(u;b).Finally, we consider the model with periodic threshold dividend strategy when the jump is exponential distributed in the fifth chapter. Similarly, the expected discounted dividend function G(u; b) is divided into G1(u; b) and G2(u; b). We can derive the intergo-differential equation of G(u; b) by same methods. Unfortunately, we can't derive the analytical solution of G(u; b).