Optimal Investment And Reinsurance Problem And The Related Game Problem In Insurance

As important financial institutions put emphasis on preventing risks in the financial market,insurance companies can help enterprises and individuals to avoid various risks and promote normal operation of the economy and society.Meanwhile,insurance companies also have the need to be more profitable.How to make rational and effective use of insurance funds has become an important topic that insurance companies have to care about.The reasonable use of insurance funds can improve insurance companies' own operating ability and solvency.Insurance companies are always faced with underwriting risks and investing risks.In order to improve their competitiveness in the financial market,insurance companies need to control these risks,and the most direct way is to purchase reinsurance.Moreover,as insurance companies play an important role in the financial market,once insurance companies do not operate well and go bankrupt,it will directly affect the normal operation of the economy and society.Therefore,it is of vitally theoretical and practical significance to study the optimal investment and reinsurance problem for insurance companies.This paper studies the optimal investment and reinsurance strategies for insurance companies from the following perspectives.(1)In Chapter 2,we study the optimal investment problem for an insurer and a reinsurer under the utility maximization criterion.The wealth process of the insurer is described by a jump-diffusion risk model.Both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset whose price process follows the constant elasticity of variance(CEV)model.Moreover,the correlation between risk model and the risky asset's price is considered.By using Legendre transform and dual theory,closedform solutions to the problems of expected exponential utility maximization are derived.(2)In Chapter 3,we consider the optimal investment and reinsurance problem for both an insurer and a reinsurer under dynamic mean-variance criterion.The wealth process of the insurer is described by the classical Cram?er-Lundberg(C-L)model.We consider both excess-of-loss reinsurance and proportional reinsurance in this chapter.Moreover,both the insurer and the reinsurer can invest in a risk-free asset and a risky asset,respectively.The price processes of risky assets are described by generalized stochastic volatility models.The special cases of this stochastic volatility model include CEV model,Heston model and GBM model.We prove the existence and uniqueness of equilibrium excess-of-loss reinsurance strategy by using the fixed point principle.In addition,The equilibrium strategies for CEV model and Heston model are also derived.(3)Different with the previous two chapters,we consider the optimal investment and reinsurance problem for both an insurer and a reinsurer under partial information with dynamic mean-variance criterion in Chapter 4.The claim process of the insurer is described by a Brownian motion with drift.The insurer and the reinsurer can invest in a risk-free asset and a risky asset,respectively.The appreciation rates of the risky assets follow O-U processes.The insurer and the reinsurer can not directly observe the appreciate rates of the risky assets,but can only observe the prices of the risky assets.By using filtering method,the problem under partial information is transformed into an equivalent problem under complete information,and the time-consistent equilibrium investment and reinsurance strategies are obtained.(4)In Chapter 5,we introduce value-added service into the optimal investment problem between two competing insurers,one provides value-added service while the other does not.Dynamic mean-variance criterion is considered in this chapter.Each insurer wants to maximize the expectation of the difference between her terminal wealth and that of her competitor,and to minimize the variance of the difference between her terminal wealth and that of her competitor.We prove the verification theorem in this chapter.By solving the corresponding extended Hamilton-Jacobi-Bellman(HJB)equations,we derive the equilibrium service level,investment strategies and the corresponding equilibrium value functions.Furthermore,numerical examples are provided to analyze the effects of parameters on the optimal strategies at the end of each chapter.