Optimal time-consistent reinsurance strategies for mean-variance insurers under thinning dependence structure

Abstract

This paper considers an optimal time-consistent proportional reinsurance problem with constraints on the strategies under the mean-variance criterion for an insurer. It is assumed that the surplus process is described by a compound Poisson risk model with dependent classes of insurance business named thinning-dependence structure, in which the stochastic sources related to claim occurrence are classified into different groups, and that each group may cause a claim in each insurance class with a certain probability. By solving the extended Hamilton-Jacobi-Bellman equation within the game theoretic framework, not only explicit expressions of the optimal strategies and value function are obtained for $n=2,$ but closed-form expressions of optimal results are derived for $n\ge 3$ by the method of dimension reduction. Moreover, some numerical examples are presented to illustrate the effects of parameters on the optimal strategies as well as the economic interpretation behind.

Keywords:

• Thinning-dependence
• time-consistent strategy
• mean-variance
• extended Hamilton–Jacobi–Bellman
• proportional reinsurance

Acknowledgements

We would like to thank the anonymous reviewers and the Editor for their constructive comments and suggestions on an earlier version of this paper, which led to a considerable improvement of the presentation of the work.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (Grant No.11471165 and Grant No.11771079) and the MOE Project of Humanities and Social Sciences (Grant No.19YJCZH083).