Paper: arXiv 2408.16989

Authors: Tim J. Boonen, Engel John C. Dela Vega

Abstract

This paper considers an insurance company that faces two key constraints: a ratcheting dividend constraint and an irreversible reinsurance constraint. The company allocates part of its reserve to pay dividends to its shareholders while strategically purchasing reinsurance for its claims. The ratcheting dividend constraint ensures that dividend cuts are prohibited at any time. The irreversible reinsurance constraint ensures that reinsurance contracts cannot be prematurely terminated or sold to external entities. The dividend rate and reinsurance level are modeled as nondecreasing processes, thereby satisfying the constraints. Claims are modeled using a Brownian risk model. The main objective is to maximize the cumulative expected discounted dividend payouts until the time of ruin. The reinsurance and dividend levels are restricted to a finite set. The optimal value function is shown to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. A threshold strategy is constructed and shown to be optimal. Finally, numerical examples are presented to illustrate the optimality conditions and optimal strategies.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical formulation using Hamilton-Jacobi-Bellman equations, viscosity solutions, and stochastic optimal control to address dynamic ratcheting and irreversible constraints. While mathematically complete and novel in its multi-constraint framework, it lacks empirical validation with real-world insurance claim datasets, relying instead on stylized numerical simulations.

Research Flowchart

  flowchart TD
    A[Research Goal: Maximize Expected Discounted Dividends<br>under Ratcheting Dividends & Irreversible Reinsurance] --> B[Model Formulation:<br>Brownian Risk Model with Nondecreasing Controls]
    B --> C[Derive Hamilton-Jacobi-Bellman<br>HJB Equation]
    C --> D[Prove Unique Viscosity Solution<br>for Optimal Value Function]
    D --> E[Construct Optimal Threshold Strategy]
    E --> F[Numerical Analysis & Verification]
    F --> G[Key Outcome: Validated Optimal Dividend<br>& Reinsurance Policy before Ruin]

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