The Variance Under Different Risk Assets Model Optimal Control Problem

As the development of insurance industry, the insurance companies are making an investment through their surplus to get more money, so as to improve the ability of compensating. At the same time, the companies themselves are reinsuring for the compensation to reduce the risk of making high payment. Any investment has it’s risk. It’s a serous question for companies to look for the optimal proportion of reinsurance and the optimal investment strategy, making the companies obtaining the desired wealth while having the least risk. The study of this kind of question really has significant meaning both theoretically and practically.The paper mainly talks about the optimal control problem of mean-variance under the consideration of that the risky assets obey CEV model, O-U model and Heston model. Aimed at improving performance of the insurance companies, taking two aspects into consideration: on the one hand, using the strategy of reinsuring for the compensation to reduce the risk; on the other hand, thinking about that the companies also have an expectation on assets appreciation, making an investment through their surplus into risk(stocks) markets to get more money, so as to get higher income. According to this two points, taking the set of strategies at moment t into consideration. The strategy set includes the proportion of reinsurance and the proportion of risk assets. So, in order to get the optimal proportion of reinsurance and the optimal investment strategy, the paper get the objective function while fixing income(expectation) and minimizing risks(variance).Firstly, using the stochastic control theory, the paper gets the Hamilton-Jacobi-Bell man(HJB) equation as the variance(risk) should get the minimum value. Secondly, using Legendre transform, Dual transform to transfer the nonlinear partial differential equation into linear partial differential equation. Thirdly, through the theory of Calculus, the paper get the derivative transform relations between value function and dual function. Finally, solving the equations to get the optimal proportion of reinsurance and the optimal investment strategy, and the explicit expression of value function. In the third chapter of the CEV model, the paper gives the relation between the optimal reinsurance and some of the factors through Numerical simulation using MATLAB and provides some support and reference to insurance practice theoretically.