The Research Of Optimal Investment Problem With Dividend And Transaction Cost

Most of the researches have only considered the investment problem for the insurance company while have ignored the management of the reinsurance com-pany recently. But the reinsurance company also faces ruin and needs to invest its asset in a financial market to manage its wealth. So, the paper mainly dis-cusses the optimal investment problem for both the insurance company and the reinsurance company when the insurance company can purchase proportional reinsurance. And we derive the relevant conclusion for the insurance company in the sense of maximizing the expected exponential utility of terminal wealth and for the reinsurance company in the sense of maximizing the expected ex-ponential utility of terminal wealth and minimizing the ruin probability. the main ideas of the thesis are as follows:In chapter one, we simply summarize the background of the insurance com-pany and the reinsurance company and their development. Then, we present the main work of this paper and the main result of my research.In chapter two, we mainly introduce several claim processes and surplus processes, the market investment price process will be used in this article.In chapter three, firstly, we model the claim process according to a Brow-nian motion with drift. Next, we consider that the insurance company can purchase proportional reinsurance to reduce the underlying risk, thus it obtains the surplus process for the insurance company. Then, we assume that the in-surance company can invest its surplus with a risk-free asset and a risky asset in a financial market and get its wealth process, the risky asset of insurance com-pany is characterized by a geometric Brownian motion. Finally, we solve the Hamilton-Jacobi-Bellman by using stochastic control theory which is satiated by the value function, and we obtain the optimal proportional reinsurance and the optimal investment strategy of maximizing expected exponential utility of terminal wealth for the insurance company.In chapter four, under the condition of the optimal proportional reinsur-ance which has been obtained in chapter three, we derive the optimal investment strategies for the reinsurance company in the sense of maximizing the expect-ed exponential utility of terminal wealth and minimizing the ruin probability respectively. Then we illustrate the equality under the two cases between max-imizing the expected exponential utility and minimizing the ruin probability about optimal investment strategies.In chapter five, we introduce the reinsurance company which can invest its surplus with a risk-free asset and a risky asset in a financial market, the risky asset is characterized by the classical constant elasticity of variance mod-el. By solving the corresponding Hamilton-Jacobi-Bellman equation and the power case which is used in the utility function, the optimal investment strate-gy is obtained.