Portfolio Optimization Strategy In Question Is Estimated

The optimal portfolio choice problem is one of the central subjects in finance, and the theory of portfolio selections is becoming more and more perfect after the development of more than fifty years. However, the analytical solutions to the optimal portfolio problem are not obtained generally in practical market. What we can do in general is to get the approximations of this problem. Therefore, it has some economic significance and academic value to evaluate the difference between the given approximate solution and the optimal solution.The duality method is used usually in the study of the optimal problem. This method changes a constrained optimal portfolio choice problem in incomplete market into the unconstrained optimal portfolio choice problem of complete markets, and gives the upper bound of the difference.There are many results to estimate the given policy by the duality method in the optimal portfolio choice problem without consumption. But those researches only give the upper bound of the difference. So the paper firstly extends the dual formulation to find the lower bound of the original constrained optimal problem.For the optimal portfolio choice with consumption, the research to estimate the policy by the duality method, so far as we know, is open now. Thereby, the paper discusses the optimal portfolio choice problem with consumption policy, and then gets the supper bound of the difference by using the similar argument. At the same time, some examples are also posed in this paper.