Continuous-Time Finite-Horizon Optimal Investment and Consumption Problems with Proportional Transaction Costs

Abstract

In this thesis, the continuous-time finite-horizon optimal investment and consumption problems with proportional transaction costs are studied. Through probabilistic approach, we investigate the optimal investment problem for a Constant Relative Risk Aversion (CRRA) investor and reveal analytically the connections between the stochastic control problem and an optimal stopping problem, with the existence of optimal stochastic controls and under certain parameter restrictions. Besides, the optimal investment and consumption problem for a Constant Absolute Risk Aversion (CARA) investor is studied through Partial Differential Equation (PDE) approach. Important analytical properties of the value function and the free boundaries for the optimal investment and consumption problem are shown through rigorous PDE arguments, while comparison is made between the two cases. In addition, the jump diffusion feature is incorporated into the optimal investment problem for a CARA investor and numerical results are provided.