The Application Of Gerber-shiu Function Under Several Risk Models

The famous actuaries Hans Gerber and Elias Shiu published the article “On the time value of ruin”in 1998, In the paper they put forward the expected discounted penalty function to study the joint distribution of laplace transform of ruin time, the instantaneous surplus before ruin and the deficit at ruin, and the function also be called the Gerber-Shiu function. With the development of the risk theory, many scholars try to research the dividend problems by Gerber-Shiu function in risk model, and they obtain many theoretical results. So the function is a risk measurement tool for analyzing the dividend and ruin problems, and also it has a very important theoretical research value.In this paper, we main research the dividend problems in several risk models and give some analysis methods of the Gerber-Shiu function. The main content of the paper is listed as follows.(1) A classical compound poisson risk model perturbed by diffusion under barrier dividend strategy is established. Integral expressions are derived for the expected present value and Gerber-Shiu functions respectively, then we can prove that they are twice continuously differentiable with respect to u.The integro-differential equations they satisfied are got according to the ?Ito ’s formula. Finally, the solution of the integro-differential equation for( )b,dfu is given in particular cases.(2) We research the Erlang(n) renewal risk model perturbed by diffusion under threshold dividend strategy, then the integro-differential equations for the moment-generation function, and Gerber-Shiu function and also the mth moment of the discounted dividend payments before ruin are deduced, in addition, the boundary conditions are obtained. At last, we will provide the special instance of the Erlang(2) risk model.(3) The compound binomial risk model extends to the case where the premium income process is no longer a linear function, which is based on a binomial process, we main research the randomized dividend strategy in this risk model. When the 0 ￡u <a and u 3a,the recursion formulas are derived for the Gerber-Shiu penalty function by using the full probability formula and dominated convergence theorem, then we can further calculate the expressions of the solutions for the renewal equations and the ultimate ruin probability in both cases, this is the main innovation of this paper.