Dividend ratcheting and capital injection under the Cramér-Lundberg model: Strong solution and optimal strategy
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arXiv
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2026
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| author | Guan, Chonghu Xu, Zuo Quan |
| author_facet | Guan, Chonghu Xu, Zuo Quan |
| contents | We consider an optimal dividend payout problem for an insurance company whose surplus follows the classical Cramér-Lundberg model. The dividend rate is subject to a ratcheting constraint (i.e., it must be nondecreasing over time), and the company may inject capital at a proportional cost to avoid ruin. This problem gives rise to a stochastic control problem with a self-path-dependent control constraint, costly capital injections, and jump-diffusion dynamics. The associated Hamilton-Jacobi-Bellman (HJB) equation is a partial integro-differential variational inequality featuring both a nonlocal integral term and a gradient constraint.
We develop a systematic probabilistic and PDE-based approach to solve this HJB equation. By discretizing the space of admissible dividend rates, we construct a sequence of approximating regime-switching systems of ordinary integro-differential equations. Through careful a priori estimates and a limiting argument, we prove the existence and uniqueness of a \emph{strong solution} in a suitable space. This regularity result is fundamental: it allows us to characterize the optimal dividend policy via a switching free boundary and to construct an explicit optimal feedback control strategy. To the best of our knowledge, this is the first complete solution -- comprising both the value function and an implementable optimal strategy -- for a dividend ratcheting problem with capital injection under the Cramér-Lundberg model. Our work advances the mathematical theory of optimal stochastic control beyond the standard viscosity solution framework, providing a rigorous foundation for dividend policy design in economics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04641 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dividend ratcheting and capital injection under the Cramér-Lundberg model: Strong solution and optimal strategy Guan, Chonghu Xu, Zuo Quan Optimization and Control Analysis of PDEs Mathematical Finance Portfolio Management 35R35, 35Q93, 91G10, 91G30, 93E20, 45K05 We consider an optimal dividend payout problem for an insurance company whose surplus follows the classical Cramér-Lundberg model. The dividend rate is subject to a ratcheting constraint (i.e., it must be nondecreasing over time), and the company may inject capital at a proportional cost to avoid ruin. This problem gives rise to a stochastic control problem with a self-path-dependent control constraint, costly capital injections, and jump-diffusion dynamics. The associated Hamilton-Jacobi-Bellman (HJB) equation is a partial integro-differential variational inequality featuring both a nonlocal integral term and a gradient constraint. We develop a systematic probabilistic and PDE-based approach to solve this HJB equation. By discretizing the space of admissible dividend rates, we construct a sequence of approximating regime-switching systems of ordinary integro-differential equations. Through careful a priori estimates and a limiting argument, we prove the existence and uniqueness of a \emph{strong solution} in a suitable space. This regularity result is fundamental: it allows us to characterize the optimal dividend policy via a switching free boundary and to construct an explicit optimal feedback control strategy. To the best of our knowledge, this is the first complete solution -- comprising both the value function and an implementable optimal strategy -- for a dividend ratcheting problem with capital injection under the Cramér-Lundberg model. Our work advances the mathematical theory of optimal stochastic control beyond the standard viscosity solution framework, providing a rigorous foundation for dividend policy design in economics. |
| title | Dividend ratcheting and capital injection under the Cramér-Lundberg model: Strong solution and optimal strategy |
| topic | Optimization and Control Analysis of PDEs Mathematical Finance Portfolio Management 35R35, 35Q93, 91G10, 91G30, 93E20, 45K05 |
| url | https://arxiv.org/abs/2604.04641 |