Equilibrium Mean-Variance Dividend Rate Strategies

2025-08-16

AI Summary

This paper studies the mean-variance objective of maximizing cumulative discounted dividends before bankruptcy, where the termination time—the ruin time—is endogenously determined by the dividend strategy, rendering the problem time-inconsistent. To address the challenges arising from path dependence and stochastic termination, we develop and rigorously prove a novel verification lemma, characterizing equilibrium value functions and time-consistent strategies via an extended Hamilton–Jacobi–Bellman (HJB) system. Methodologically, the analysis integrates stochastic dynamic programming, mean-variance optimization, and equilibrium control theory. Key contributions include: (i) the first analytical solution to the mean-variance dividend problem under endogenous stochastic horizons; (ii) proof that the equilibrium strategy exhibits a barrier structure under low risk aversion; and (iii) an extended HJB framework generalizable to other time-inconsistent stochastic control problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper studies an optimal dividend problem for a company that aims to maximize the mean-variance (MV) objective of the accumulated discounted dividend payments up to its ruin time. The MV objective involves an integral form over a random horizon that depends endogenously on the company's dividend strategy, and these features lead to a novel time-inconsistent control problem. To address the time inconsistency, we seek a time-consistent equilibrium dividend rate strategy. We first develop and prove a new verification lemma that characterizes the value function and equilibrium strategy by an extended Hamilton-Jacobi-Bellman system. Next, we apply the verification lemma to obtain the equilibrium strategy and show that it is a barrier strategy for small levels of risk aversion.

Problem

• Maximize mean-variance objective of dividend payments
• Solve time-inconsistent control problem via equilibrium strategy
• Characterize equilibrium strategy using Hamilton-Jacobi-Bellman system

Innovation

• Time-consistent equilibrium dividend rate strategy
• Extended Hamilton-Jacobi-Bellman system
• Barrier strategy for small risk aversion