A Study On Optimal Premiums And Optimal Investment-reinsurance Problems Of Insurance Companies

As a financial institution,insurance companies usually charge premiums to prepare for years of losses in the future.Therefore,it is the most important point faced by every insurance company to formulate a reasonable premium policy.In addition,insurance companies must find ways to maintain profitability in order to spread risk and increase the value of the company's assets.For most insurance companies,investment and reinsurance are two important ways which insurance companies often choose.This thesis studies the optimal premium strategy of insurance companies under a stochastic interest rate economy and the optimal investment-reinsurance strategy under model uncertainty.It is composed of seven chapters as follows:In the first chapter,we mainly introduce the research background and literature review,preliminary knowledge and model introduction,the main work as well as innovation and deficiencies.The second chapter is concerned with the optimization problem with stochastic interest rates faced by an insurance firm who can control its cash-balance dynamics by adjusting the underlying premium rate.The firm's objective is to minimize the total deviation of its cash-balance process.Thus we obtain the optimal premium policy and the optimal cost function.In the third and the fourth chapter,we study the optimization problems for meanvariance insurers,where the surplus process of the insurer is assumed to follow a jumpdiffusion model.By using the stochastic dynamic programming approach and solving the Hamilton-Jacobi-Bellman(HJB)equations,we obtain the optimal investment(and reinsurance)strategies and the value function,respectively.In the fifth chapter,we study optimal investment-reinsurance strategies for an insurer who faces model uncertainty.The insurer is allowed to acquire new business and invest into a financial market which consists of one risk-free asset and one risky asset whose price process is modeled by a Geometric Brownian motion.Minimizing the expected quadratic distance of the terminal wealth to a given benchmark under the“worst-case”scenario,we obtain the closed-form expressions of optimal strategies and the corresponding value function by solving the HJB equation.Numerical examples are presented to show the impact of model parameters on the optimal strategies.The sixth chapter considers a robust optimal portfolio selection problem of an insurer whose surplus is modeled by a jump-diffusion risk process.The financial market consists of one risk-free asset and one risky asset whose price process satisfies the Heston model.By solving a Hamilton-Jacobi-Bellman-Isaacs(HJBI)equation,we derive the explicit expressions of the optimal portfolio process and the corresponding value function.Furthermore,a verification result and some technical conditions for a welldefined value function are provided.Numerical examples are presented to show the impact of model parameters on the optimal strategies.In the last chapter,we investigate a robust optimal reinsurance and investment problem for an Ambiguity-Averse Insurer(AAI),whose surplus is described by the classical compound Poisson(or Cram(?)r-Lundberg)risk process.Suppose that the AAI can purchase proportional reinsurance and invest into a financial market consisting of one risk-free asset and one risky asset whose price process satisfies Heston model.By applying the stochastic control approach,we derive the closed-form expressions of the candidate optimal strategies and the candidate value function,and then we provide convenient sufficient conditions for the verification theorem.Numerical examples are presented to show the impact of model parameters on the optimal strategies.