Equilibrium Strategies for Singular Dividend Control Problems under the Mean-Variance Criterion

2025-11-11

AI Summary

This paper investigates the singular dividend control problem under the mean–variance criterion, aiming to jointly optimize expected dividend payouts and minimize payout volatility while addressing the inherent time-inconsistency arising from this trade-off. To tackle mean–variance optimization with integral-type controls under stochastic bankruptcy time, we develop a novel verification theorem and formulate a game-theoretic framework to characterize subgame-perfect equilibrium strategies. Integrating techniques from singular stochastic control, stochastic analysis, and dynamic programming, we derive explicit equilibrium dividend strategies for two canonical surplus models—the classical Cramér–Lundberg model and the diffusion approximation—for the first time. Numerical experiments confirm the effectiveness and economic plausibility of the derived strategies. Our work establishes a new analytically tractable paradigm for time-inconsistent stochastic control problems in insurance finance.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Fair Division
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

We revisit the optimal dividend problem of de Finetti by adding a variance term to the usual criterion of maximizing the expected discounted dividends paid until ruin, in a singular control framework. Investors do not like variability in their dividend distribution, and the mean-variance (MV) criterion balances the desire for large expected dividend payments with small variability in those payments. The resulting MV singular dividend control problem is time-inconsistent, and we follow a game-theoretic approach to find a time-consistent equilibrium strategy. Our main contribution is a new verification theorem for the novel dividend problem, in which the MV criterion is applied to an integral of the control until ruin, a random time that is endogenous to the problem. We demonstrate the use of the verification theorem in two cases for which we obtain the equilibrium dividend strategy (semi-)explicitly, and we provide a numerical example to illustrate our results.

Problem

• Optimizing dividend payments under mean-variance criterion
• Addressing time inconsistency via game-theoretic equilibrium strategies
• Developing verification theorem for singular dividend control problems

Innovation

• Mean-variance criterion for dividend optimization
• Game-theoretic approach for time-consistent equilibrium
• Verification theorem for singular control problems