| The dual risk model has been research widely both at home and abroad in recent years, and the observation is continuous in these models. However, this assumption is too restrictive in applications. In order to give a more reasonable description of the actual behavior of the company, randomized observation are introduced to the risk model. In this article, we consider the dual risk model with interest rates and jumps under random observation,and study relevant problems of bankruptcy. In addition,we discuss the Markov-modulated risk model under multi-layer dividend strategy and obtained some results of the Gerber-Shiu discounted penalty function. The main structure of the paper is as follows:In Chapter1, we simply introduce the background of the risk theory and its development.In Chapter2, elementary basis of knowledge and some risk models are presented briefly.In Chapter3, on the basis of the dual risk model with interest,we take randomized observation into consideration,obtain a new model. A system of integro-differential equation with certain boundary conditions satisfied by the Gerber-Shiu discounted penalty function are derived. It is difficult to find the explicit solution, so we use the Sine methods to derive an approximate solution.In Chapter4, we discuss the dual risk model with two-sided jumps un-der randomized observation, and derive a system of integro-differential equa-tion with certain boundary conditions satisfied by the Gerber-Shiu discounted penalty function. We also use the Sine methods to derive an approximate solu-tion.In Chapter5, we study a regime-switching risk model with multi-layer dividend strategy. We assume that the claim amount is regulated by an external Markov chain, the integro-differential and integral equations for Gerber-Shiu expected discounted penalty function are derived. At last, we consider the case where the claim amount distribution Fi belongs to the rational family, explicit solutions are derived by Laplace transforms. |
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