A deep learning-driven iterative scheme for high-dimensional HJB equations in portfolio selection with exogenous and endogenous costs

Abstract: In this paper, we first conduct a study of the portfolio selection problem, incorporating both exogenous (proportional) and endogenous (resulting from liquidity risk, characterized by a stochastic process) transaction costs through the utility-based approach. We also consider the intrinsic relationship between these two types of costs. To address the associated nonlinear two-dimensional Hamilton-Jacobi-Bellman (HJB) equation, we propose an innovative deep learning-driven policy iteration scheme with three key advantages: i) it has the potential to address the curse of dimensionality; ii) it is adaptable to problems involving high-dimensional control spaces; iii) it eliminates truncation errors. The numerical analysis of the proposed scheme, including convergence analysis in a general setting, is also discussed. To illustrate the impact of these two types of transaction costs on portfolio choice, we conduct through numerical experiments using three typical utility functions.