Paper: arXiv 2511.19701

Authors: Paulin Aubert, Etienne Chevalier, Vathana Ly Vath

Abstract

In this paper, we study an optimal dividend and capital-injection problem in a Cramér–Lundberg model where claim arrivals follow a Hawkes process, capturing clustering effects often observed in insurance portfolios. We establish key analytical properties of the value function and characterise the optimal capital-injection strategy through an explicit threshold. We also show that the value function is the unique viscosity solution of the associated HJB variational inequality. For numerical purposes, we first compute a benchmark solution via a monotone finite-difference scheme with Howard’s policy iteration. We then develop a reinforcement learning approach based on policy-gradient and actor-critic methods. The learned strategies closely match the PDE benchmark and remain stable across initial conditions. The results highlight the relevance of policy-gradient techniques for dividend optimisation under self-exciting claim dynamics and point toward scalable methods for higher-dimensional extensions.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical formulation of optimal dividend and capital injection strategies under self-exciting Hawkes processes, complete with viscosity solution proofs and dynamic programming principles. However, its empirical rigor is limited to synthetic numerical PDE benchmarks and reinforcement learning simulations rather than real-world data backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Optimal Dividend & Capital Injection<br>under Hawkes Claim Clustering"] --> B["Mathematical Formulation:<br>Cramér-Lundberg Model + Hawkes Process"]
    B --> C["Analytical Characterization:<br>HJB Variational Inequality &<br>Threshold Capital Injection"]
    C --> D["Numerical PDE Benchmark:<br>Monotone Finite-Difference Scheme<br>+ Howard's Policy Iteration"]
    C --> E["Reinforcement Learning:<br>Policy-Gradient & Actor-Critic Methods"]
    D --> F["Key Outcomes:<br>• Unique Viscosity Solution Established<br>• RL Strategies Closely Match PDE Benchmark<br>• Scalable Framework for High-Dim Problems"]
    E --> F

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3 of 160 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.