A Study Of Several Types Of Optimal Dividends, Reinsurance And Investment Portfolio Problems

In the field of actuarial science,the optimal stochastic control problem has always been a hot issue,which includes optimal dividend,optimal reinsurance and optimal investment problem.The optimal dividend problem is studied under the classical risk model,by adding an exponential penalty function allows for the case of negative surplus.According to whether the dividend rate is bounded,the optimal combination of dividend and reinsurance is discussed,in which the reinsurance premium is calculated by the standard deviation principle.Considering that insurance companies may have incomplete or missing information about the financial market,which leads to probability uncertainty,the optimal investment strategy of an ambiguity averse insurance company is studied by adding ambiguity to the model.In practical application situation,the time consistency of the strategy is an important and reasonable requirement,and this paper also studies the time-consistent optimal investment reinsurance strategy with ambiguity aversion under the mean-variance criterion.This paper uses the dynamic programming principle to establish the Hamilton-Jacobi-Bellman(HJB)equation for the corresponding problem,and finally derives the expressions of the value functions and gives the corresponding optimal strategies,and finally describes the practical implications by numerical examples.In this paper,the optimal control of the combinatorial problem in different cases is discussed.The first chapter introduces the popular research problems in the actuarial field of insurance.The current status of domestic and international research in recent years is analyzed,and relevant classic papers are presented.The second chapter investigates the problem of optimal dividend strategy with an exponential penalty function under the classical risk model.The objective is to maximize the discounted value of expected dividends before bankruptcy,allowing for the case of negative surplus,which is expressed by incorporating an exponential form of the penalty function.Expressions for the value function on three intervals are given,and the corresponding optimal dividend strategy is described.The third chapter investigates the problem of optimal dividend and proportional reinsurance strategies under the standard deviation premium principle.The objective is to maximize the expected discounted value of dividends.Based on the value of the dividend rate,the cases of bounded and unbounded dividends are discussed,and the expressions of the value functions are obtained for the two cases,respectively.The fourth chapter investigates the optimal investment strategy problem with ambiguity aversion under the exponential premium principle.The objective is to maximize the expected utility value of the terminal wealth,assuming that the insurance company is ambiguity averse.Considering ambiguity in financial markets,the alternative model is described by relative entropy,and the premium rate is described by the exponential premium principle.The expression of the value function of optimal investment reinsurance is deduced.The fifth chapter investigates the problem of a time-consistent robust investment reinsurance strategy with ambiguity aversion under the mean-variance criterion.The objective is to find a time-consistent strategy that maximizes the expectation of terminal wealth in the worst case,while minimizing the variance of its terminal wealth.The corresponding HJB equation is developed and the expressions of the value function are derived.