Paper: arXiv 2610.13162

Authors: Xinagyu Cui, Xun Li, Ziyan Li, Yun Shi

Abstract

This paper studies entropy regularization for the discrete-time dynamic mean–variance portfolio problem with no-shorting constraints. In the unconstrained setting, entropy regularization can be introduced without changing the basic quadratic structure of the dynamic program. Under no-shorting, this property is lost: a standard fixed-temperature formulation generally preserves convexity but no longer preserves the homogeneous Bellman structure and the associated piecewise-linear feedback form. We introduce a state-dependent temperature that is matched to the homogeneity of the original mean–variance problem. This restores the piecewise-quadratic value function and the finite-dimensional Bellman recursion. The resulting formulation, however, makes convexity depend on the continuation coefficients generated by the soft recursion. We show that negative continuation coefficients lead to loss of convexity and prove that a common sufficiently small temperature preserves the Bellman structure and convexity over the entire finite horizon. The original no-shorting solution is recovered as the temperature vanishes. We also develop a sample-based implementation for unknown return distributions. Numerical and empirical results illustrate both the structural advantage of the state-dependent temperature and the practical relevance of the convexity-preserving temperature restriction.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly mathematical treatment of entropy regularization in dynamic mean-variance optimization, introducing a novel state-dependent temperature to preserve structure. While primarily theoretical, it includes numerical and empirical results, demonstrating a solid level of empirical rigor in validating the proposed methodology. The combination of advanced mathematical derivations and practical application places it in the ‘Holy Grail’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Address entropy regularization for dynamic mean-variance portfolio with no-shorting] --> B{Methodology: Entropy Regularization & State-Dependent Temperature};
    B --> C[Problem Formulation: Standard vs. State-Dependent Temperature];
    C --> D{Analysis & Derivations: Bellman Equations, Convexity, Structure Preservation};
    D --> E[Computational Processes: Numerical Simulations & Empirical Studies];
    E --> F{Key Outcomes: State-Dependent Temperature Restores Structure & Convexity under Sufficiently Small Temperature};
    F --> G[Further Outcomes: Original No-Shorting Solution Recovered as Temperature Vanishes; Sample-based Implementation Developed];