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

2025-09-02

AI Summary

This paper addresses the portfolio selection problem under combined exogenous (proportional) and endogenous (liquidity-risk-driven stochastic) transaction costs. The resulting high-dimensional, nonlinear two-dimensional Hamilton–Jacobi–Bellman (HJB) equation poses severe computational challenges. To overcome this, we propose a deep learning–driven policy iteration method that integrates deep neural networks, stochastic differential equation modeling, and a truncation-free policy iteration framework—thereby circumventing both the curse of dimensionality and numerical truncation errors while enabling efficient solution of high-dimensional stochastic control problems. Numerical experiments across three canonical utility functions demonstrate the algorithm’s convergence and robustness. Importantly, our approach provides the first quantitative characterization of the coupling mechanism between the two types of transaction costs and their joint impact on optimal investment policies.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Solving high-dimensional HJB equations for portfolio selection
• Incorporating exogenous and endogenous transaction costs
• Addressing curse of dimensionality with deep learning

Innovation

• Deep learning-driven policy iteration scheme
• Addresses curse of dimensionality
• Eliminates truncation errors