Paper: arXiv 2604.04641

Authors: Chonghu Guan, Zuo Quan Xu

Abstract

We consider an optimal dividend payout problem for an insurance company whose surplus follows the classical Cramér-Lundberg model. The dividend rate is subject to a ratcheting constraint (i.e., it must be nondecreasing over time), and the company may inject capital at a proportional cost to avoid ruin. This problem gives rise to a stochastic control problem with a self-path-dependent control constraint, costly capital injections, and jump-diffusion dynamics. The associated Hamilton-Jacobi-Bellman (HJB) equation is a partial integro-differential variational inequality featuring both a nonlocal integral term and a gradient constraint. We develop a systematic probabilistic and PDE-based approach to solve this HJB equation. By discretizing the space of admissible dividend rates, we construct a sequence of approximating regime-switching systems of ordinary integro-differential equations. Through careful a priori estimates and a limiting argument, we prove the existence and uniqueness of a \emph{strong solution} in a suitable space. This regularity result is fundamental: it allows us to characterize the optimal dividend policy via a switching free boundary and to construct an explicit optimal feedback control strategy. To the best of our knowledge, this is the first complete solution – comprising both the value function and an implementable optimal strategy – for a dividend ratcheting problem with capital injection under the Cramér-Lundberg model. Our work advances the mathematical theory of optimal stochastic control beyond the standard viscosity solution framework, providing a rigorous foundation for dividend policy design in economics.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical contribution to stochastic control theory using Hamilton-Jacobi-Bellman equations and PDE analysis under the Cramér-Lundberg model. While mathematically complex and highly novel in establishing strong solutions and optimal policies, it lacks empirical validation, backtesting, or empirical datasets.

Research Flowchart

  flowchart TD
    A["Research Goal: Optimal Dividend Strategy & Capital Injection<br><i>(Cramér-Lundberg Model with Ratcheting Constraint)</i>"] --> B["Problem Formulation: Stochastic Control & HJB Equation<br><i>(Integro-Differential Variational Inequality)</i>"]
    
    B --> C["Space Discretization of Dividend Rates<br><i>(Approximating Regime-Switching Systems)</i>"]
    
    C --> D["Analysis & Convergence Arguments<br><i>(A Priori Estimates & Limiting Argument)</i>"]
    
    D --> E["Proving Strong Solution Existence & Uniqueness<br><i>(PDE Regularity Result)</i>"]
    
    E --> F["Key Outcome: Characterize Switching Free Boundary &<br>Construct Optimal Feedback Control Strategy"]