Optimal Reinsurance And Investment Based On The CRRA Utility Risk Model

The combination of reinsurance and investment of insurance companies has always been a hot issue in insurance business.In insurance practice,it is an inevitable choice for insur-ance companies to invest their own funds in order to expand the company's capital scale and enhance market competitiveness in order to obtain greater returns.At the same time,in order to spread the insurance risks and avoid the inability to fulfill the liability for compen-sation due to over-concentration of claims or occurrence of catastrophe.Buying reinsurance business has also become an important approach.Therefore,it is an important research top-ic for insurance companies to effectively control risk investment and reinsurance strategies,and to maximize the utility of expected wealth or minimize the probability of bankruptcy.Stochastic control theory has also been widely applied in the research of reinsurance and investment issues.In this paper,a diffusion risk model is used to describe the surplus of an insurance company.By purchasing proportional reinsurance,the amount of claims to be borne is reduced to avoid bankruptcy caused by excessive claims.Meanwhile,assets are put into the classic Black-Scholes financial market,which includes a risk investment and a risk-free investment.A target value function of V(x)isdefined.This paper studies the problem under the preference assumption of CRRA utility,which plays a more important role in the insurance companies to reasonably arrange their financial investment,disperse risks and attract customer resources.The structure of this article is as follows:The introduction part of the first chapter introduces the research background and cur-rent situation of insurance companies on the optimal control of investment and purchase reinsurance strategies,as well as the research content of the article.The second chapter introduces the preparatory knowledge and risk model used in this paper,and defines the objective function V(x).The third chapter mainly introduces the main results of this paper.Firstly,using the dynamic programming principle in the random control theory,we obtain HJB equation with dividend utility of v(x)satisfied,prove the existence of the equation solution of v(x)and study the properties of satisfying the equation solution and obtain that the target value function of V(x)is a monotonically increasing concave function.In this paper,we discuss the initial capital of x and the initial capital of x respectively.The critical value of x*for initial funds is found,and the optimal reinsurance and investment strategy under CRRA utility and the expression of the target value function of V(x)are obtained,with the form of V(x)as follows:The optimal expected dividend utility of c*is c*(x)=(V(x))-/1-p,The expression of the optimal investment strategy pi ast that maximizes the expected dividend utility,Expression of optimal reinsurance policy q*The fourth chapter summarizes the conclusions of the model studied in this paper and looks forward to the future development direction in this area.