Monte Carlo Scheme for a Singular Control Problem

Nowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To modelling these financial problems in the real world, the qualitative and quantitative behaviors of the solution to a free boundary problem are still not well understood and also numerical solutions to free boundary problems remain a challenge. Stochastic control problems reduce to free-boundary problems in partial differential equations while there are no bounds on the rate of control. In a free boundary problem, the solution as well as the domain to the PDE need to be determined simultaneously. In this dissertation, we concern the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite time portfolio selection problem with proportional transaction costs. We consider optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption. The problem is mainly governed by a time-dependent Hamilton-Jacobi-Bellman equation with gradient constraints. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the high-dimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013).