Comparative Analysis of Deep Galerkin Method and Finite Difference Method for Solving PDEs in Portfolio Optimization

Abstract

Since its inception by Markowitz, mean-variance analyses have played a significant role in portfolio management, helping investors make wise and rational decisions. However, the mean variance has faced major drawbacks due to its inability to incorporate factors such as taxes and transaction costs. This has led to the framework being improved in many ways. One of the well-known models for portfolio optimization that improves upon mean-variance analyses is the famous Merton portfolio problem. Merton himself solved the problem, utilizing the dynamical programming approach, which transformed the problem into a partial differential equation framework. It is worth noting that the resulting PDE generally lacks analytical solutions. In such cases, traditional numerical methods are employed to obtain the numerical solutions of the PDE. However, several studies have indicated that, at higher dimensions, these numerical methods present a significant computational challenge and tend to be slow. This issue is commonly referred to as the curse of dimensionality. To deal with the curse of dimensionality, deep learning algorithms are now being used. This study aims to investigate the performance of the deep learning algorithm, the Deep Galerkin method (DGM), in comparison to the finite difference method (FDM). At first, both the mean-variance analysis framework and the Merton problem framework are presented. To solve the Merton problem, the HJB equation is utilized to transform the problem into the nonlinear partial differential equation (PDE) and the associated optimal controls. Furthermore, the resulting PDE and optimal control are solved by implementing Python code for both the DGM and FDM. The results demonstrate that, in general, the former outperforms the latter. We also observed that, at various time points, DGM consistently provided more accurate results compared to FDM.