# American Institute of Mathematical Sciences

March  2021, 17(2): 841-868. doi: 10.3934/jimo.2020001

## Optimal stop-loss reinsurance with joint utility constraints

 1 Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai 200241, China 2 Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia 3 Department of Financial Engineering, Ningbo University, 818 Fenghua Road, Ningbo 315211, China

* Corresponding author: Wei Wang

Received  May 2018 Revised  August 2019 Published  March 2021 Early access  January 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (11771147, 11571113), the Zhejiang Provincial Natural Science Foundation of China (LY17G010003), "Shuguang Program" supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission(18SG25), the Humanity and Social Science Youth Foundation of the Ministry of Education of China (18YJC910012), the National Social Science Foundation Key Program (17ZDA091), the 111 Project(B14019) and Faculty Research Grant of University of Melbourne

We investigate the optimal reinsurance problems in this paper, specifically, the stop-loss strategies that can bring mutual benefit to both the insurance company and the reinsurance company. The utility improvement constraints are adopted by both contracting parties to guarantee that a reinsurance contract will bring higher expected utilities of wealth to the two participants. We also introduce five risk criteria that reflect the interests of both parties. Under each optimality criterion, we obtain explicit expressions of optimal stop-loss retentions and the corresponding optimised value of objective functions. The upper and lower bounds of expected utility increments under the optimal stop-loss retentions are provided. In the numerical example, we analyse the expected utility improvements under the criterion of minimising total Value-at-Risk. Notable increases in the lower bound of total utility increments are observed after adopting the joint utility improvement constraints.

Citation: Nan Zhang, Linyi Qian, Zhuo Jin, Wei Wang. Optimal stop-loss reinsurance with joint utility constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 841-868. doi: 10.3934/jimo.2020001
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##### References:
The expected utility increment without utility constraint when $\alpha_I<\alpha_R$
The expected utility increment with utility constraints when $\alpha_I<\alpha_R$
The lower bound of total expected utility increments when $\alpha_I<\alpha_R$
The expected utility increment without utility constraint when $\alpha_I>\alpha_R$
The expected utility increment with utility constraints when $\alpha_I>\alpha_R$
The lower bound of total expected utility increment when $\alpha_I>\alpha_R$
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