Stochastic Optimal Control Method for Portfolio Selection

It is of great importance to study portfolio selection with stochastic interest rates and liability for both theoretical analysis and practical applications. In this paper, on maximizing the expected utility of terminal wealth, the interest rate is supposed to be satisfied with the Vasicek process, and the optimal investment problem with liability and fixed proportion transaction cost is established. HJB equation of the value function of the optimal investment problem with liability is obtained by the maximum principle, and nonlinear HJB equation is tranformed into linear PDE by Legendre transform-dual method. Furthermore, the closed-form solutions of the optimal investment strategy is derivied under the logarithmic utility function.