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

Paper: arXiv 2511.08433

Authors: Jingyi Cao, Dongchen Li, Virginia R. Young, Bin Zou

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical formulation of a time-inconsistent singular dividend control problem using mean-variance criteria and game theory. While highly advanced mathematically with rigorous proofs and custom verification theorems, it lacks empirical validation or backtesting on real-world financial data, relying instead on a stylized numerical simulation.

Research Flowchart

  flowchart TD
    A[Research Goal: Find time-consistent dividend strategies<br/>under Mean-Variance criterion in singular control] --> B[Data & Inputs: De Finetti model with<br/>diffusion cash reservoir & endogenous ruin time]
    B --> C[Methodology: Game-theoretic approach<br/>for time-inconsistent problem]
    C --> D[Develop Novel Verification Theorem<br/>for MV criterion on integral control until ruin]
    D --> E[Apply Theorem to Solve 2 Cases<br/>Semi-explicitly + Numerical Example]
    E --> F[Key Outcome: Equilibrium dividend strategy balancing<br/>high expected payout with low variability]