Paper: arXiv 2508.08130

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

Abstract

This paper considers an insurer with two collaborating business lines, and the risk exposure of each line follows a diffusion risk model. The manager of the insurer makes three decisions for each line: (i) dividend payout, (ii) (proportional) reinsurance coverage, and (iii) capital injection (from one line into the other). The manager seeks an optimal dividend, reinsurance, and capital injection strategy to maximize the expected weighted sum of the total dividend payments until the first ruin. We completely solve this problem and obtain the value function and optimal strategies in closed form. We show that the optimal dividend strategy is a threshold strategy, and the more important line always has a lower threshold to pay dividends. The optimal proportion of risk ceded to the reinsurer is decreasing with respect to the aggregate reserve level for each line, and capital injection is only used to prevent the ruin of a business line. Finally, numerical examples are presented to illustrate the impact of model parameters on the optimal strategies.

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 framework using stochastic control and diffusion processes to derive closed-form optimal dividend, reinsurance, and capital injection strategies. While the mathematical derivation is advanced and complete, empirical testing is restricted to basic numerical examples rather than real-world market data.

Research Flowchart

  flowchart TD
    A[Goal: Maximize expected weighted dividends for 2 collaborating business lines] --> B[Model Formulation: Diffusion Risk Model with Dividend, Reinsurance & Capital Injection]
    B --> C[Methodology: Stochastic Control Theory & Hamilton-Jacobi-Bellman Equations]
    C --> D[Derive Closed-Form Value Function & Optimal Strategies]
    D --> E[Numerical Analysis & Parameter Sensitivity Testing]
    E --> F[Key Outcomes: Threshold Dividends, Reserve-Dependent Reinsurance & Capital Injection Strategy]