Two Risk Models With A Hybrid Dividend Strategy

There are two parts in this paper, we focus on two risk models with a hybrid dividend strategy.Dividend strategies for insurance risk model were first proposed by De Finetti in1957. From then on, more and more scholars paid attention to the risk model with dividend strategy in the field of actuarial science. As a very important task in the study of insurance and finance, it has become one of the hot topics in the current actuarial science research. Ornstein-Uhlenbeck-type model, which is one of the significant models has recently gained a lot of attentions. Ornstein-Uhlenbeck model with Barrier dividend strategy and threshold dividend strategy have been considered. In the first part of this paper, we define a new kind of dividend strategy to Ornstein-Uhlenbeck model, namely the hybrid dividend strategy. We define hybrid dividend strategy as follows:the company pays dividends according to the following strategy governed by parameters b2> b1>0, and α>0; whenever the modified surplus is below the level b1, no dividends are paid; however, when the modified surplus is above b1and below the b2, dividends are paid continuously at a constant rate α; when the modified surplus is above b2, dividends are complete paid. According to the above dividend strategy, we first introduce the model and find the expression of dividend functions until ruin. Secondly, we consider the limit of dividend level, and comparing with the known results. Thirdly, we consider Laplace transform of ruin time and get its expression. Finally, the partial differential equations with boundary conditions satisfied by the moments and moment-generating function are proved.The second part of this paper, we consider a more general problem of dividend for one-dimensional diffusion process. Firstly, we introduce the one-dimensional diffusion process. Secondly, we apply Ito formula to prove the expression of Laplace transform and boundary conditions for the first passage time. According to the theorem we consider several popular diffusion processes. Thirdly, based on the previous chapter we study the hybrid dividend strategy for the general diffusion process and obtain more specific results. Finally, some applications in the calculation of dividend value functions are considered.