Optimal dividend and capital injection under self-exciting claims

Paulin Aubert, Etienne Chevalier and Vathana Ly Vath

Papers from arXiv.org

Abstract: In this paper, we study an optimal dividend and capital-injection problem in a Cram\'er--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.

Date: 2025-11
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