Study On HJB Equation Of Investment Portfolio And Its Viscosity Solutions

In this paper, we consider various influence factors of investment and consumption activities in reality, by using stochastic optimal control theory, the viscosity solution theory and Lie symmetry method, we study the following several portfolio and consumption problems under the framework of continuous-time, obtain the corresponding HJB equation, prove properties of the problems'value functions, and get the optimal strategies, consequently reveal the real meaning of its applications.This paper's main work and results are mainly in chapter 3, 4, 5.In chapter 3, we consider the optimal life insurance purchase, consumption and portfolio problem for the investors under an uncertain lifetime in the continuous time. By using the dynamic programming principle, we get the HJB equation on life insurance purchase, consumption and investment strategies, and prove the continuity and concavity of the value function. Finally with the help of the viscosity solutions theory, we prove that the value function is the only viscosity solution of corresponding HJB equation.In chapter 4, we study the optimal portfolio selection problem which risky assets follow geometric Brownian motion. With the help of the dynamic programming principle and Lie algebra theory, we get the corresponding HJB equation, obtain the analytical expressions of the value function based on Lie symmetry, and present the optimal investment strategy. In the end, the undetermined coefficient method shows the validity of the result.In chapter 5, we research the investment portfolio model with proportional transaction costs. Firstly, we analyze the finance market model and the dynamic equation, according to the utility function we establish the index function and the corresponding optimization model. And then with the help of the dynamic programming principle, we get the differential equation satisfied the value function, and analyze the properties of the value function, reduce the determined problem to the free boundary problem. Finally, we discuss the numerical method of solving the free boundary problem.