Extended HJB Equation for Mean-Variance Stopping Problem: Vanishing Regularization Method

Paper: arXiv 2510.24128

Authors: Yuchao Dong, Harry Zheng

Abstract

This paper studies the time-inconsistent MV optimal stopping problem via a game-theoretic approach to find equilibrium strategies. To overcome the mathematical intractability of direct equilibrium analysis, we propose a vanishing regularization method: first, we introduce an entropy-based regularization term to the MV objective, modeling mixed-strategy stopping times using the intensity of a Cox process. For this regularized problem, we derive a coupled extended Hamilton-Jacobi-Bellman (HJB) equation system, prove a verification theorem linking its solutions to equilibrium intensities, and establish the existence of classical solutions for small time horizons via a contraction mapping argument. By letting the regularization term tend to zero, we formally recover a system of parabolic variational inequalities that characterizes equilibrium stopping times for the original MV problem. This system includes an additional key quadratic term–a distinction from classical optimal stopping, where stopping conditions depend only on comparing the value function to the instantaneous reward.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using an entropy-regularized Cox process to derive an extended HJB equation for time-inconsistent mean-variance stopping problems. While highly sophisticated mathematically with novel theoretical limits, it is purely theoretical and completely lacks empirical backtesting or data applications.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Time-Inconsistent<br/>Mean-Variance Optimal Stopping Problem] --> B[Data/Inputs: Reward Function,<br/>State Process & Regularization Parameter ε]
    B --> C[Step 1: Introduce Entropy Regularization<br/>via Cox Process Intensity]
    C --> D[Step 2: Derive & Solve Coupled<br/>Extended HJB System via Contraction Mapping]
    D --> E[Step 3: Apply Vanishing Regularization<br/>Method as ε → 0]
    E --> F[Key Outcome: Parabolic Variational Inequality System<br/>with New Quadratic Term for Equilibrium Stopping]

Cited References (19)

1. On time-inconsistent stochastic control in continuous time (2017)
2. On time-inconsistent stopping problems and mixed strategy stopping times (2020)
3. Randomized optimal stopping problem in continuous time and reinforcement learning algorithm (2024)
4. Time consistent stopping for the mean-standard deviation problem—The discrete time case (2019)
5. Second order parabolic differential equations (1996)
6. Optimal stopping and free-boundary problems (2006)
7. Formulae for the derivatives of heat semigroups (1994)
8. Randomized optimal stopping algorithms and their convergence analysis (2021)

0 of 19 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.