JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (11): 1-14,26.doi: 10.6040/j.issn.1671-9352.0.2022.265

    Next Articles

Optimal reinsurance contract design based on Stackelberg game

Peng YANG1(),Xiaoyan ZHANG2   

  1. 1. School of Mathematics, Xi'an University of Finance and Economics, Xi'an 710100, Shaanxi, China
    2. Xi'an Changfeng Research Institute of Mechanism and Electricity, Xi'an 710065, Shaanxi, China
  • Received:2022-04-29 Online:2023-11-20 Published:2023-11-07

RichHTML

10

PDF (PC)

465

Abstract:

For the optimal reinsurance contract design problem, that is, the reinsurance premium pricing problem, we assume that there is an insurance company and a reinsurance company on the market. Among them, the insurance company is engaged in two types of insurance business, and the reinsurance company is engaged in one type of insurance business. To reflect the competition between the insurance company and the reinsurance company, we assume that these three insurance businesses are interdependent, and the insurance company can engage in the reinsurance business. Using the relative performance, the competition between the insurance company and the reinsurance company is quantified. The aims of the insurance company and the reinsurance company are to find an optimal reinsurance contract to maximize the mean of the terminal wealth and minimize the variance of the terminal wealth. Under the Stackelberg game framework, by using stochastic calculus and stochastic control theory, the explicit solutions for the optimal reinsurance contract and the optimal value function are obtained. Finally, the influence of model parameters on the optimal reinsurance contract is analyzed through numerical experiments and the relationship between some special cases and general cases is compared.

Key words: reinsurance contract, reinsurance premium, insurance business dependence, relative performance, Stackelberg game

CLC Number: 

  • O211.6

Fig.1

Effects of whether the reinsurance company is engaged in insurance business and τ1 on the optimal reinsurance contract"

Fig.2

Effects of whether the reinsurance company is engaged in insurance business and τ2 on the optimal reinsurance contract"

Fig.3

Effects of whether the reinsurance company is engaged in insurance business and λ on the optimal reinsurance contract"

Fig.4

Effects of whether the dependence between insurance businesses is considered and λ on the optimal reinsurance contract"

Fig.5

Effects of whether the insurance company is considered competition and λ on the optimal reinsurance contract"

1 RONG Ximin , YAN Yiqi , ZHAO Hui . Asymptotic solution of optimal reinsurance and investment problem with correlation risk for an insurer under the CEV model[J]. International Journal of Control, 2023, 96 (4): 840- 853.
doi: 10.1080/00207179.2021.2015627
2 ZHANG Yan , ZHAO Peibiao , TENG Xinghua , et al. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform[J]. Journal of Industrial and Management Optimization, 2021, 17 (4): 2139- 2159.
doi: 10.3934/jimo.2020062
3 CHEN Zhiping , YANG Peng . Robust optimal reinsurance-investment strategy with price jumps and correlated claims[J]. Insurance: Mathematics and Economics, 2020, 92, 27- 46.
doi: 10.1016/j.insmatheco.2020.03.001
4 LI Zhongfei , ZENG Yan , LAI Yongzeng . Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV models[J]. Insurance: Mathematics and Economics, 2012, 51 (1): 191- 203.
doi: 10.1016/j.insmatheco.2011.09.002
5 YANG Peng , CHEN Zhiping , WANG Liyuan . Time-consistent reinsurance and investment strategy combining quota-share and excess of loss for mean-variance insurers with jump-diffusion price process[J]. Communications in Statistics: Theory and Methods, 2021, 50 (11): 2546- 2568.
doi: 10.1080/03610926.2019.1670849
6 LIN Xiang , QIAN Yiping . Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model[J]. Scandinavian Actuarial Journal, 2016, 2016 (7): 646- 671.
doi: 10.1080/03461238.2015.1048710
7 YU Wenguang , GUO Peng , WANG Qi , et al. On a periodic capital injection and barrier dividend strategy in the compound Poisson risk model[J]. Mathematics, 2020, 8 (4): 511.
doi: 10.3390/math8040511
8 EISENBERG J , MISHURA Y . Optimising dividends and consumption under an exponential CIR as a discount factor[J]. Mathematical Methods of Operations Research, 2020, 92, 285- 309.
doi: 10.1007/s00186-020-00714-w
9 HAN X , LIANG Z B , YOUNG V R . Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle[J]. Scandinavian Actuarial Journal, 2020, 2020 (10): 879- 903.
doi: 10.1080/03461238.2020.1788136
10 LUESAMAI A . Lower and upper bounds of the ultimate ruin probability in a discrete time risk model with proportional reinsurance and investment[J]. The Journal of Risk Management and Insurance, 2021, 25 (1): 1- 10.
11 YUAN Yu , LIANG Zhibin , HAN Xia . Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs[J]. Journal of Industrial and Management Optimization, 2022, 18 (2): 933- 967.
doi: 10.3934/jimo.2021003
12 PERCY D F , WARNER D B . Evaluating relative performances in disabled sports competitions[J]. IMA Journal of Management Mathematics, 2009, 20, 185- 199.
13 DE MARZO P M , KANIEL R , KREMER I . Relative wealth concerns and financial bubbles[J]. The Review of Financial and Studies, 2008, 21 (1): 19- 50.
doi: 10.1093/rfs/hhm032
14 DENG Chao , ZENG Xudong , ZHU Huiming . Non-zero-sum stochastic differential reinsurance and investment games with default risk[J]. European Journal of Operational Research, 2018, 264 (3): 1144- 1158.
doi: 10.1016/j.ejor.2017.06.065
15 PUN C S , SIU C C , WONG H Y . Non-zero-sum reinsurance games subject to ambiguous correlations[J]. Operations Research Letters, 2016, 44 (5): 578- 586.
doi: 10.1016/j.orl.2016.06.004
16 HU Duni , WANG Hailong . Time-consistent investment and reinsurance under relative performance concerns[J]. Communications in Statistics: Theory and Methods, 2018, 47 (7): 1693- 1717.
doi: 10.1080/03610926.2017.1324987
17 CHEN Shumin , YANG Hailiang , ZENG Yan . Stochastic differential games between two insurers with generalized mean-variance premium principle[J]. ASTIN Bulletin, 2018, 48 (1): 413- 434.
doi: 10.1017/asb.2017.35
18 WANG Ning , ZHANG Nan , JIN Zhuo , et al. Stochastic differential investment and reinsurance games with nonlinear risk processes and VaR constraints[J]. Insurance: Mathematics and Economics, 2021, 96, 168- 184.
doi: 10.1016/j.insmatheco.2020.11.004
19 GU Ailing , CHEN Shumin , LI Zhongfei , et al. Optimal reinsurance pricing with ambiguity aversion and relative performance concerns in the principal-agent model[J]. Scandinavian Actuarial Journal, 2022, 2022 (9): 749- 774.
doi: 10.1080/03461238.2022.2026459
20 YANG Peng , CHEN Zhiping . Optimal reinsurance pricing, risk sharing and investment strategies in a joint reinsurer-insurer framework[J]. IMA Journal of Management Mathematics, 2022, 34 (4): 661- 694.
21 BÄUERLE N . Benchmark and mean-variance problems for insurers[J]. Mathematical Methods of Operations Research, 2005, 62 (1): 159- 165.
doi: 10.1007/s00186-005-0446-1
22 BJÖRK T , MURGOCI A . A general theory of Markovian time inconsistent stochastic control problems[J]. SSRN Electronic Journal, 2010, 1, 55.
23 YANG Peng , CHEN Zhiping , XU Ying . Time-consistent equilibrium reinsurance-investment strategy for n competitive insurers under a new interaction mechanism and a general investment framework[J]. Journal of Computational and Applied Mathematics, 2020, 374, 112769.
doi: 10.1016/j.cam.2020.112769
[1] ZHOU Chi, WANG Yi-xin, YU Jing. Wholesale, agency or hybrid? sales mode choices of hybrid platform in competition with self-run platform [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(1): 89-100.
[2] XIAO Min, YU Min, HE Xin-hua. Channel pricing strategy of supply chain considering vertical E-commerce [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(3): 110-118.
[3] CHENG Xi, WANG Chuan-xu, XU Lang. Decision-making in low-carbon supply chain considering altruism preference [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(5): 41-52.
[4] ZHANG Ke-yong1, WU Yan1, HOU Shi-wang2. Pricing strategy of considering retailer’s fairness concerns in  the closed-loop supply chain [J]. J4, 2013, 48(05): 83-91.
[5] LU Yuan-yuan. Cooperative advertising strategy and order policy under retailers’ competition [J]. J4, 2011, 46(1): 71-78.
[6] LU Yuan-yuan , . Strategic multi-brand retailer response to the asymmetric supply chain channel [J]. J4, 2008, 43(8): 19-23 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] MA Tong-yi1,2, LIU Chun-yan2. The generalized Shephard problem for Lp-mixed projection bodies and Minkowski-Funk transforms[J]. J4, 2012, 47(10): 21 -30 .
[2] SUN Lin, ZHENG Yu, YAO Juan. Robust non-fragile H∞ state feedback control for  uncertainty discrete singular systems[J]. J4, 2011, 46(1): 35 -41 .
[3] . Joint test in nonlinear regression models with AR(2) random errors[J]. J4, 2009, 44(7): 38 -43 .
[4] QIN Zhao-yu,LIU Shi-lian*,YANG Yin-rong,LIU Fu-jun,LI Jian-yuan,SONG Chun-hua . Technology exploration for proteomics analysis in hepatopancreas of shrimp (Fenneropenaeus chinensis) with white spot syndrome[J]. J4, 2007, 42(7): 5 -08 .
[5] WU Dai-yong. Global attractivity of a nonlinear discrete Logistic model  with feedback control[J]. J4, 2013, 48(4): 114 -110 .
[6] LIU Hua ,LIU Zhi-guang ,SU Min ,LI Zi-zhen , . Clumping effect in a host—parasitoid interaction model[J]. J4, 2008, 43(8): 31 -34 .
[7] . Asymptotic behavior of  nonoscillatory solutions of second  order nonlinear differential equation with perturbation[J]. J4, 2009, 44(2): 19 -23 .
[8] ZHANG Fang-guo. Elliptic curves in cryptography: past, present and future…[J]. J4, 2013, 48(05): 1 -13 .
[9]

GAO Li-li,MA Xia,ZHAO Hong-fei,PEI Jia-wei,L〖AKU¨〗 Zhao-lin,ZHANG Bo-lin *

.

Physiological characteristics of Lactobacillus delbrueckii subsp. lactis
mutant strains with reduced H+-ATPase activity

[J]. J4, 2008, 43(7): 1 -09 .
[10] YU Wen-guang1, HUANG Yu-juan2. Ruin probability in an interfered multi-type risk model of a variable ruin limit[J]. J4, 2011, 46(3): 58 -62 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd