Optimal Dividend Problem In A Piecewise Deterministic Compound Poisson Risk Model

With the progress of the times,people's insurance awareness is gradually improving,naturally promoting the development of the insurance industry and even the financial field.At the same time,the research on insurance theory has also attracted great interest and attention from many scholars.One of their main concerns is to assess and improve the stability of insurance companies as far as possible.Through the establishment of an actuarial mathematical model to simulate the company's operating process,they use mathematical tools such as stochastic process theory,martingale theory and probability statistics to analyze and research the model.The results of the research on the ruin probability and its related actuarial variables are used to assess the risk control ability of insurance companies.From another relatively straightforward realistic point of view,if an insurance company always pays dividends to shareholders before bankruptcy,the amount of dividends can reflect the company's economic strength.Then,the realistic problem can be transformed into the optimization problem in the insurance risk model.The mathematical tool used in the study of the dividend problem is the stochastic control theory,and the objective function is expanded based on the maximum of the sum of the expected cumulative discounted dividend payments.The research on dividend strategy focuses on barrier strategy,threshold strategy,linear dividend barrier,and band strategy.In recent years,with the gradual integration of insurance and finance,the issue of optimal dividends has increasingly attracted the attention of researchers.They are constantly making perfect of the insurance risk model,and they are also focusing on the use of new mathematical tools and methods to study different dividend strategies.On the basis of the predecessors,this paper proceeds from the piecewise deterministic compound Poisson risk model,and studies the problem of restricted dividends,unrestricted dividends,and optimal dividends with perturbed diffusion factors.Compared with the general risk model,premium income per unit time for this model depends on the company's surplus process,which is more in line with our reality:when the company's surplus lies in a higher level,the insurance company can strengthen its market by reducing premiums.and when the surplus lies in a lower level,insurance company can increase premium income to avoid the risk of underfunding.In addition,the risk model also covers many types of risk models that appear in literatures.The Gerber-Shiu discount penalty function is also considered in the construction of the objective function,which is more in line with the actual situation and has a certain research significance.Based on piecewise-deterministic compound Poisson risk model,the paper research several dividend problems and the basic framework of the article is as follows:In first chapter,the introduction presents the background of risk theory and dividend strategy,and briefly reviews the domestic and foreign research results related to the content of the article.At the same time,it briefly describes the main content of the article.In second chapter,according to the read literature,the preparatory knowledge in the dividend strategy study is described,including the Cramer-Lundberg classical risk model,the barrier dividend strategy,the threshold dividend strategy,the linear dividend barrier,and the important risk measurement tool Gerber-Shiu Discount penalty functions.In third chapter,based on piecewise-deterministic compound Poisson risk model,this section research an optimal dividend assignment problem with regard to an insurance company.Our objective is to maximize the sum of the mean of the cumulative discounted dividend payments until ruin and a penalty payment at the time of ruin.Then in the case of bounded dividend rate,we present explicitly corresponding Hamilton-Jacobi-Bellman equation which satisfied the optimal value function.Furthermore,the value function is demonstrated to be a solution of the associated HJB equation by verification theorem,and it is shown that the optimal dividend strategy is the threshold strategy.When the dividend rate is unconstrained,the optimal dividend is the barrier dividend strategy.In fourth chapter,we further introduce the perturbed diffusion factor to piecewise-deterministic compound Poisson risk model.At the same time,we discuss the optimal dividend problem under the model from the perspective of linear dividend barrier,and the objective function is the expected discount of the accumulated dividend amount.And the explicit expression of value function is derived under the linear dividend barrier condition when the claim size follows the exponential distribution.In fifth chapter,we summarize the research results of the paper and considers where the text can be studied in depth.