Paper: arXiv 2509.02267

Authors: Dong Yan, Nanyi Zhang, Junyi Guo

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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical derivation of a high-dimensional nonlinear HJB equation incorporating both exogenous and endogenous (stochastic liquidity) transaction costs. It introduces an innovative deep learning-driven policy iteration scheme (using PINNs) and provides formal convergence analysis, though empirical evaluation relies on synthetic numerical experiments rather than real market backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve High-Dim HJB Equations for Portfolio Selection with Exogenous & Endogenous Costs] --> B[Inputs: Exogenous/Proportional Costs, Endogenous Liquidity Costs & 3 Utility Functions]
    B --> C[Formulate Nonlinear 2D HJB Equation]
    C --> D[Deep Learning-Driven Policy Iteration Scheme]
    D --> E[Computational Processes: Neural Network Policy Updates & Convergence Analysis]
    E --> F[Key Outcomes: Overcomes Curse of Dimensionality, Eliminates Truncation Errors & Quantifies Cost Impacts]

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0 of 39 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.