A Numerical Scheme for A Singular control problem: Investment-Consumption Under Proportional Transaction Costs

TL;DR

This paper develops a numerical scheme combining Monte Carlo simulation and finite difference methods to solve a complex, high-dimensional Hamilton-Jacobi-Bellman equation for optimal investment and consumption with transaction costs.

Contribution

It introduces a novel numerical approach for solving a nonlinear parabolic double obstacle problem in portfolio optimization with transaction costs.

Findings

Numerical results reveal optimal trading strategies consistent with theoretical properties.

The method effectively handles high-dimensional problems.

The approach accurately approximates the value function gradients.

Abstract

This paper concerns the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite portfolio selection with proportional transaction costs. We consider the 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).