Title
Extended HJB Equation for Mean-Variance Stopping Problem: Vanishing Regularization Method
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.
Bio
Yuchao Dong is an Assistance Professor from School of Mathematical Sciences, Tongji University. His research focus on stochastic optimal control theory and its applications in mathematical finance, especially on the theory of continuous time reinforcement learning. His works have been published on journals including SIAM Journal on Control & Optimization, SIAM Journal on Applied Mathematics, SIAM Journal on Mathmatical Analysis and Mathmatical Finance. He got his PhD from School of Mathematical Sciences, Fudan University.