Optimal dividends for regulated insurers with a nonlinear penalty

Abstract

This study investigates an optimal dividend problem for an insurer with random time regulation and a nonlinear penalty from the payment of dividends. The surplus process of the insurer is a jump–diffusion model and the goal of the insurer is to maximise the expected accumulated discount dividend payment until ruin occurs or regulatory time arrives. The introduction of the external random regulatory time leads that the HJB equation associated with this control problem is a time in-homogeneous parabolic variational inequality. There is no common explicit solution to this variational inequality. We derived explicit solutions for optimal policies when both the regulatory time and the claims follow an exponential distribution. For general cases, this paper provides a Markov chain approximation method for the value function and optimal policies and presents an example to illustrate the algorithm when the random regulatory time follows the Gompertz–Makeham distribution.

Keywords:

• Optimal dividends
• solvency regulation
• dynamic programming
• Markov chain approximation
• Gompertz–Makeham distribution

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Peng Xu and Zhenlong Chen were supported by the National Natural Science Foundation of China [Grant No. 11971432], Natural Science Foundation of Zhejiang Province [Grant No. LY21G010003], First Class Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics), and Collaborative Innovation Center of Statistical Data Engineering Technology & Application. Lin Xu was supported by the Natural Science Foundation of China [Grant No. 11971034] and Humanities and Social Sciences project of the Ministry of Education of China [Grant No. 17YJC910009].