Optimal Investment and Reinsurance Problems under Mean-Variance and Game Framework

© 2021 Jiannan Zhanghis thesis studies several optimal investment problems in a dynamic environment from different perspectives. Under different settings, we derive explicit expressions of the optimal functions and the corresponding objective strategies. Firstly, we consider a continuous-time mean-variance portfolio selection problem based on a log-return model. The aim is to find the optimal investment proportion that achieves a trade-off between the mean and variance of return. The optimal allocation is dependent on the wealth level. Asset-liability management is also an interesting topic in the field of finance. In the third chapter, we formulate a mean-variance with the consideration of a liability process. Liability is a natural control variable and we use debt ratio to control the liability. After applying forward backward stochastic differential equations (FBSDEs), we obtain closed-form solutions of the optimal investment strategies and the corresponding efficient frontiers, respectively. Since the financial market is full of uncertainty and the insurance industry consists of a variety of institutions, the insurer should consider the strategic interaction when determining strategies. We implement the game theory in a reinsurance-investment problem to reflect the cooperation and competition among insurers. We focus on the time-consistent non-zero-sum reinsurance-investment stochastic differential game problem between two insurers under mean-variance criteria with state-dependent risk aversions. Plugging a series of FBSDEs, we derive the optimal reinsurance proportion and investment policies. In addition, dividend payment strategy is also an important issue in financial and actuarial studies. In the forth chapter, we investigate a class of non-zero-sum stochastic differential game problems between two insurance companies with dividend. Both companies have the option of paying dividends. The optimization of the dividend payment in models is described as a singular stochastic control. Applying the theory of the stopping game into the study of singular control problem, we can connect the singular control problem to a stopping game. Adopting this connection, we derive the optimal dividend payment policies for two insurance companies. Finally, based on the study findings, some conclusions and studies of further research are made