Optimal investment problems over a finite time horizon

Abstract

This thesis is concerned with optimal investment problems over a finite time horizon. The value function is constructed and the corresponding Hamilton-Jacobi-Bellman (HJB) equation can be derived by applying dynamic programming. In this thesis, we derive the properties of the strategy as well as the boundary and terminal line. We also discuss the optimal stopping time with multi-assets. The main contents of this thesis are divided into three parts. In the first part, we study an optimal consumption investment model with uncertain exit time. The value function is not only the expectation of utility of the price of assets on maturity date, but also the expected utility produced in the whole process. Using the method of partial differential equation (PDE), we prove the smoothness of the value function without specifying a particular utility function, where a non-smooth and non-concave situation is considered. Some restrictions are imposed on the problem. The continuity of the optimal strategy and some properties of the boundary and terminal line are derived. In the second part, we discuss the above problem with constraints. The value function can be characterized by two types of second-order partial differential equations in different regions. One is a fully nonlinear equation, and the other is a linear equation. We construct an approximation problem to make the equations satisfy the parabolic condition. Using the method of partial differential equation, we prove the existence, uniqueness and regularity of the solution to the original problem via the approximation problem. We derive the properties of the free boundary line and ascertain its end point. In the third part, we consider the optimal stopping time for investors to leave the financial market among multi-assets to obtain maximum profit. The utility function is considered as a quadratic form. Two models are researched respectively in this part. One is with a normal utility function, and the other is based on a Logarithmic utility-maximization objective. A two-stage problem is formulated. The main problem is a nonstandard optimal stopping time problem. Using the method of stochastic analysis, we turn it into a standard one. The subproblem with control variable in the drift and volatility terms is solved via stochastic control method. Numerical examples are also presented accordingly to illustrate the efficiency of the theoretical results.