To Study The Dynamic Investment Selection Problem With Short-selling Restrictions

Optimal portfolio selection is one of the central issues in the theory and practice of modern mathematical finance. Nowadays, it is widely recognized that the asset-liability management is of both theoretical interest and practical importance. It is also find that liability and no-shorting constrain are two important factors which almost all investors should cope with. It is meaningful that the introduction of liability and no-shorting constrain in a portfolio selection problem.In this paper, we give a systematic study in continuous-time mean-variance portfolio selection problem with liability in the case where short-selling the stocks is not allowed. Firstly, we study a portfolio selection that is formulated as a bi-objective optimization problem. The objective is to maximize the expected terminal return and minimize the variance of the terminal wealth. By using the Karush-Kuhn-Tucher conditions, we get the closed form of the optimal portfolio strategy and the efficient frontier. Then, we study a asset-liability model while the liability evolves according to a Brownian motion with drift and jump. The goal is to minimize the variance of the terminal wealth subject to a given expected terminal return. We also obtain the optimal portfolio strategy and the efficient frontier in closed forms. Finally, we study the mean-variance portfolio selection problem under the stock with jump with the liability and no-shorting constrains. By using HJB equation and viscosity theory, we get the explicit solution of the optimal portfolio strategy and the efficient frontier.