On The Generalized Negative Risk Models

Risk theory is an important part of the actuarial mathematics and analyzes the various risk models with stochastic mathematics as a major tool. Ruin probability is the core of risk theory. Ruin probability provides a reference to the company's management and decision-making to guide the healthy development of the risk model.According to the different claims, the risk models can be divided into positive risk model and negative risk model. Most of the existing literature is to study the positive risk model. With the development of insurance and risk theory, negative risk model also attracted strong interest of many scholars in recent years. Domestic researchers of the negative risk model are more and more, such as Liu Zaiming, Wang Yunjie, Xiong Shuang[30-32] and so on. Some foreign scholars mainly study the negative risk model with dividend problems, such as [36-39]. Based on compound Poisson process, the negative risk model, considering the different "claims" means and dividend strategy, has established the Erlang(n) negative risk model, and the expected discounted dividend function was studied.In this thesis, the work as following:In Chapter2, on the basis of the classic negative risk model under compound Poisson process, we consider the number of "claims" to promote a composite of two random sequences and the random interference, and establish the negative risk model with aggregate claims modeled as two compound negative binomial processes. We derive the ruin probability and the Lundberg inequality.In Chapter3, considering the different claims and dividends, the Erlang(n) negative risk model under a threshold dividend strategy is studied, but also the expected discounted dividend payments to satisfy an integro-differential equations. The resolution of the equation in the case of exponential distribution expression is derived.