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Local Discontinuous Galerkin Method for Portfolio Optimization with Transaction Costs

We study the continuous time portfolio selection problem over a finite horizon for an investor who maximizes the expected utility of terminal wealth and faces transaction costs. The portfolio consists of a risk-free asset, and a risky asset whose price is modeled as a geometric Brownian motion. The problem can be formulated as a stochastic singular control or an impulse control problem depending on whether the transaction costs are of proportional or fixed type. Due to the intractability of the problem, modelers resort to numerical methods to obtain approximations of solutions to the problem. In this paper we propose a stable and high-order computational scheme to solve this problem, which is capable of handling any form of transaction costs. Specifically, we implement the Local Discontinuous Galerkin (LDG) Finite Element Method (FEM) to solve the resulting convection-diffusion Partial Differential Equation (PDE), and obtain error estimates for the LDG method. Moreover, we prove the convergence of the scheme. Our numerical experiments show the order of accuracy of the LDG method and illustrate the optimal policies under various kinds of transaction costs.