Two free boundary problems in optimal investment

Abstract

We have considered two free boundary problems in optimal investment. Problem I is concerned with the optimal decision to sell or buy a stock in a given period with reference to the ultimate average of the stock price. This is an optimal stopping time problem which can be formulated as a variational inequality problem. We provide a partial differential equation (PDE) approach to study the optimal strategy. Problem II concerns numerical solutions for the continuous-time portfolio selection with proportional transaction costs which is described as a singular stochastic control problem. The associated value function is governed by a variational inequality with gradient constraints. We propose a penalty method to deal with the gradient constraints and then employ the finite difference discretization.