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Optimal dividend and capital injection under self-exciting claims

Paulin Aubert, Etienne Chevalier, Vathana Ly Vath

TL;DR

The paper tackles the problem of optimally distributing dividends and injecting capital in a surplus model where claim arrivals are self-exciting via a Hawkes process. It combines analytical results—establishing bounds, monotonicity, a capital-injection threshold, and a viscosity-solution characterization of the HJB variational inequality—with numerical PDE benchmarks and reinforcement-learning approaches (policy gradient and actor-critic) to learn near-optimal strategies. The key contributions include an explicit injection threshold $oldsymbol{ abla^ op}$, a demonstration that the value function is the unique viscosity solution, and a scalable RL framework that matches PDE benchmarks and remains stable across initial conditions. The practical impact lies in providing a tractable, data-driven method to navigate dividend and capital decisions under clustered claim dynamics, with potential extension to higher-dimensional settings.

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.

Optimal dividend and capital injection under self-exciting claims

TL;DR

The paper tackles the problem of optimally distributing dividends and injecting capital in a surplus model where claim arrivals are self-exciting via a Hawkes process. It combines analytical results—establishing bounds, monotonicity, a capital-injection threshold, and a viscosity-solution characterization of the HJB variational inequality—with numerical PDE benchmarks and reinforcement-learning approaches (policy gradient and actor-critic) to learn near-optimal strategies. The key contributions include an explicit injection threshold , a demonstration that the value function is the unique viscosity solution, and a scalable RL framework that matches PDE benchmarks and remains stable across initial conditions. The practical impact lies in providing a tractable, data-driven method to navigate dividend and capital decisions under clustered claim dynamics, with potential extension to higher-dimensional settings.

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.

Paper Structure

This paper contains 25 sections, 14 theorems, 76 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Modelling insurer's portfolio and clustering effect
  3. Uncontrolled surplus dynamics
  4. Temporal claim dependence via a Hawkes process
  5. Controlled surplus dynamics
  6. Ruin and objective function
  7. Theoretical analysis
  8. Pre-claim intensity
  9. Dynamic programming principle
  10. Analytical properties of the value function
  11. Capital injection strategies
  12. Hamilton--Jacobi--Bellman equation
  13. Viscosity solution characterization
  14. Finite-difference estimate
  15. Discrete HJB variational inequality
  16. ...and 10 more sections

Key Result

Proposition 3.1

Let $\theta$ be any $\mathcal{F}$-stopping time and $(x,y)\in\mathbb R\times\lbrack b,+\infty)$, it follows from the dynamic programming principle that

Figures (6)

  • Figure 1: Estimated value function and corresponding optimal control policy under the baseline parameter configuration.
  • Figure 2: Sensitivity of optimal policy to Hawkes dynamics parameters.
  • Figure 3: Sensitivity of the optimal policy to insurance parameters.
  • Figure 4: Sensitivity of the optimal policy to financing and valuation parameters.
  • Figure 5: Convergence of the learned objective toward the PDE benchmark value.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Definition 2.1: Set of admissible strategies
  • Remark 2.1
  • Proposition 3.1: Dynamic Programming Principle
  • Proposition 3.2: Value function boundaries
  • proof
  • Proposition 3.3: Monotonicity in $x$
  • proof
  • Proposition 3.4: Monotonicity in $y$
  • proof
  • Corollary 3.1
  • ...and 26 more