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

Battle Count: 86

The paper provides a rigorous theoretical foundation for mean-variance optimal stopping problems, which are directly relevant to quantitative trading decisions such as optimal timing for asset sales, portfolio liquidation, and risk-aware trade execution. The game-theoretic equilibrium framework addresses time-inconsistency, a critical issue in practical trading where risk preferences change over time. However, the paper is purely theoretical with no numerical implementation, making direct application to trading systems non-trivial. The key insight—that stopping conditions involve an additional quadratic term γ/2(f-g)² beyond the classical value-vs-reward comparison—has implications for designing risk-aware stopping rules in trading algorithms.

The paper involves highly complex mathematical machinery: coupled systems of parabolic PDEs, variational inequalities, Cox process modeling, entropy regularization, contraction mapping arguments, Itô-Tanaka formula, and game-theoretic equilibrium analysis. Implementing the theoretical results would require solving coupled nonlinear PDE systems with free boundaries, which is computationally very challenging. No numerical methods or algorithms are provided. The mathematical sophistication is at the level of advanced stochastic analysis and PDE theory.

This is a purely theoretical mathematics paper with no code, no numerical experiments, and no datasets. All results are analytical proofs (verification theorem, existence via contraction mapping, formal convergence). Reproducibility is limited to verifying the mathematical derivations. No computational implementation is provided.