Extended HJB Equation for Mean-Variance Stopping Problem: Vanishing Regularization Method
Yuchao Dong, Harry Zheng
TL;DR
This work addresses time-inconsistent mean-variance stopping and introduces a vanishing entropy regularization that models mixed stopping via a Cox-process intensity. The regularized problem yields a coupled extended HJB system with a verifiability condition linking the optimal intensity to the value function; existence is established for small horizons by a contraction argument, and sending the regularization parameter to zero formally recovers a parabolic variational-inequality description of MV equilibrium stopping. The approach provides a rigorous PDE-based framework for MV stopping, explains the appearance of a quadratic stopping term, and extends to infinite-horizon and discrete-time settings, while outlining future work on convergence proofs and numerical methods.
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.