Optimal dividend and capital injection under self-exciting claims

TL;DR

This paper investigates optimal dividend and capital injection strategies in an insurance model with claim arrivals modeled by a Hawkes process, combining analytical, numerical, and reinforcement learning methods.

Contribution

It introduces a novel approach to dividend and capital-injection optimization under self-exciting claims, including explicit strategies and scalable reinforcement learning techniques.

Findings

Optimal strategies characterized by explicit thresholds.

Reinforcement learning methods closely match PDE benchmark solutions.

Strategies are stable across different initial conditions.

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.