The Maximum Severity Of Ruin In A Generalized Erlang(N) Risk Process Perturbed By Diffusion

With the rapid development of the financial sector and the social economy, the insurance industry has been rapidly development. It attracted more experts and scholars to explore special research in this area. Most number of experts is committed to study the probability of ruin, the time of ruin, the surplus immediately before ruin and other issues. The regenerative capacity of insurance company remains to study. When the first time of the surplus of an insurance company is being the level of negative, then that is ruin. In fact, it doesn’t mean ruin truly. Insurance companies can be run through a variety of ways, such as loans. This lets it possible to make the level of the surplus restored to positive. However, the insurance companies cannot return to profitability even by the way of loans when the insurance company’s negative surplus reaches a certain value, then that is absolutely ruin. Meanwhile, the dividends are also a very important issue. The shareholders invest funds in business, get the dividend expires. So, it plays also very important to study the insurance company’s dividend under the different models and strategies.In this paper, we begin to establish a generalized Erlang(n) risk process perturbed by diffusion. Then, with the help of Laplace transform, we get an integro-differential equation that is satisfied by the survival probability. Its solution can be expressed as a linear combination of 2n linearly independent particular solutions of the integro-differential equation. When n=2, the maximum severity of ruin can be expressed explicitly in terms of the non-ruin probability. Finally, we illustrate the results of this paper through an example.The paper further discusses the introduction of a constant interest in the above risk model under a dividend barrier strategy. Under such a strategy, Integro-differential equations for the moment generating functions and moment functions of the present value of all dividends paid until ruin are derived. At the same time, the Integro-differential equations for the moment generating functions and the Gerber-Shiu function are derived by the same way. Some special cases for V1(u,b) are considered in details. Then we give a numerical solution and a detailed explanation of affecting the factors of dividends.Finally, we make a summary of the results in this paper and give a prospect of this study.