Optimal Dividend, Reinsurance, and Capital Injection for Collaborating Business Lines under Model Uncertainty

Paper: arXiv 2603.25350

Authors: Tim J. Boonen, Engel John C. Dela Vega, Len Patrick Dominic M. Garces

Abstract

This paper considers an insurer with two collaborating business lines that faces three critical decisions: (1) dividend payout, (2) reinsurance coverage, and (3) capital injection between the lines, in the presence of model uncertainty. The insurer considers the reference model to be an approximation of the true model, and each line has its own robustness preference. The reserve level of each line is modeled using a diffusion process. The objective is to obtain a robust strategy that maximizes the expected weighted sum of discounted dividends until the first ruin time, while incorporating a penalty term for the distortion between the reference and alternative models in the worst-case scenario. We completely solve this problem and obtain the value function and optimal (equilibrium) strategies in closed form. We show that the optimal dividend-capital injection strategy is a barrier strategy. The optimal proportion of risk ceded to the reinsurer and the deviation of the worst-case model from the reference model are decreasing with respect to the aggregate reserve level. Finally, numerical examples are presented to show the impact of the model parameters and ambiguity aversion 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 provides a rigorous mathematical derivation of optimal control strategies under model uncertainty using stochastic calculus and differential games, achieving closed-form solutions. However, it lacks empirical testing on real-world market data, relying instead on stylized numerical simulations.

Research Flowchart

  flowchart TD
    A[Goal: Maximize discounted dividends via robust optimal dividend, reinsurance, and capital injection under model uncertainty] --> B[Inputs: Diffusion reserve processes, ambiguity aversion parameters, reference model]
    B --> C[Method: Formulate robust stochastic control problem with worst-case distortion penalty]
    C --> D[Compute: Solve Hamilton-Jacobi-Bellman-Isaacs equations to obtain closed-form value function]
    D --> E[Finding 1: Optimal dividend & capital injection follow a barrier strategy]
    D --> F[Finding 2: Ceded risk proportion & model deviation decrease as aggregate reserve rises]

Cited References (32)

1. Optimal Dividend, Reinsurance, and Capital Injection Strategies for an Insurer with Two Collaborating Business Lines (2025) — link
2. Optimal risk exposure and dividend payout policies under model uncertainty (2021)
3. Reinsurance games with two reinsurers: Tree versus chain (2023)
4. Optimization of the flow of dividends (1995)
5. Controlling risk exposure and dividends payout schemes: insurance company example (1999)
6. Optimal dividend strategies of two collaborating businesses in the diffusion approximation model (2018)
7. Continuous-time Stochastic Control and Optimization with Financial Applications (2009)
8. Stochastic Differential Equations: An Introduction with Applications (2013)

1 of 32 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.