🤖 AI Summary
This paper addresses the time-inconsistent mean-variance optimal stopping problem, where standard dynamic programming fails to characterize equilibrium strategies. To overcome this, we propose a vanishing entropy regularization: a small entropy parameter is introduced to formulate a regularized game, yielding an extended Hamilton–Jacobi–Bellman (HJB) system featuring quadratic terms; taking the vanishing-parameter limit yields a parabolic variational inequality system that rigorously characterizes the equilibrium stopping intensity of the original problem. Theoretically, we establish existence of classical solutions on short time intervals and prove a verification theorem ensuring one-to-one correspondence between solutions and equilibrium strategies. Methodologically, we integrate Cox process modeling, contraction mapping arguments, and variational analysis to construct a rigorous convergence path from the regularized to the original problem. This framework provides the first mathematically tractable characterization of time inconsistency in mean-variance stopping problems.
📝 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.