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doi: 10.3934/jimo.2022099
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Stochastic differential game strategies in the presence of reinsurance and dividend payout

1. 

Department of Mathematics and Applied Mathematics, University of Limpopo, Private Bag X1106, Sovenga, 0727, South Africa

2. 

Department of Mathematics and Computer Science, Great Zimbabwe University, P.O. Box 1235, Masvingo, Zimbabwe

3. 

Department of Mathematics and Applied Mathematics, University of Pretoria, 0002, South Africa

4. 

Department of Mathematics and Informatics, Eduardo Mondlane University, 257, Maputo, Mozambique

*Corresponding author: Farai Julius Mhlanga

Received  February 2022 Revised  May 2022 Early access June 2022

This paper presents and examines a problem in which two insurance companies apply non-proportional reinsurance to control risk. Additionally, each firm pays out dividends. The situation is modelled as a zero-sum stochastic differential game between the two companies. The goal of one company is to maintain business competitive advantage over the other by sustaining or increasing the difference between the respective liquid reserves of the two companies while the second company aims to minimise that difference. A verification theorem is formulated, proved and subsequently employed to derive the saddle point components. For the case of the payoff with a non-zero running cost function, we are able to solve explicitly the differential game. Numerical simulations are presented to illustrate the results as well as the economic interpretation.

Citation: Farai Julius Mhlanga, Lesiba Charles Galane, Nicholas Mwareya, Eriyoti Chikodza, Calisto Guambe. Stochastic differential game strategies in the presence of reinsurance and dividend payout. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022099
References:
[1]

S. Asmussen and H. Albrecher, Ruin probabilities, 2$^{nd}$ edition World Scientific, Singapore, 2010. doi: 10.1142/9789814282536.

[2]

P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the cramér-lundberg model, Mathematical Finance, 15 (2005), 261-308.  doi: 10.1111/j.0960-1627.2005.00220.x.

[3]

N. Ba$\ddot{\text{u}}$erle, Approximation of optimal reinsurance and dividend payout policies, Mathematical Finance, 14 (2004), 99-113.  doi: 10.1111/j.0960-1627.2004.00183.x.

[4]

B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, in Transactions of the XVth International Congress of Actuaries, 2, (1957), 433–443.

[5]

D. LiX. Rong and H. Zhao, Stochastic differential game formulation on the reinsurance and investment problem, International Journal of Control, 88 (2015), 1861-1877.  doi: 10.1080/00207179.2015.1022797.

[6]

S. Luo, A stochastic differential game for quadratic-linear diffusion processes, Advances in Applied Probability, 48 (2016), 1161-1182.  doi: 10.1017/apr.2016.69.

[7]

S. Luo, M. Wang and W. Zhu, Stochastic differential reinsurance game in diffusion approximation models, Journal of Computational and Applied Mathematics, 386, (2021). doi: 10.1016/j.cam.2020.113252.

[8]

H. Meng, T. K. Siu and H. Yang, A note on optimal insurance risk control with multiple reinsurers, Journal of Computational and Applied Mathematics, 319, (2017), 38–42. doi: 10.1016/j.cam.2016.12.034.

[9]

H. Meng, T. K. Siu and H. Yang, Optimal insurance risk control with multiple reinsurers, Journal of Computational and Applied Mathematics, 306, (2016), 40–52. doi: 10.1016/j.cam.2016.04.005.

[10]

M. Taskar and X. Zeng, Optimal non-proportional reinsurance control and stochastic differential games, Insurance: Matematics and Economics, 48, (2011), 64–71. doi: 10.1016/j.insmatheco.2010.09.006.

[11]

N. Wang, N. Zhang and L. Qian, Reinsurance-investment game between two mean-variance insurers under model uncertainty, Journal of Computational and Applied Mathematics, 382, (2021). doi: 10.1016/j.cam.2020.113095.

[12]

W. Liu and Y.-J. Hu, Optimal proportional reinsurance and dividend payments with transaction costs and internal competition, Applied Mathematics-A Journal of Chinese Universities, 31 (2016), 89-102.  doi: 10.1007/s11766-016-3281-7.

[13]

J. Xu and M. Zhou, Optimal risk control and dividend distribution policies for a diffusion model with terminal value, Mathematical and Computer Modelling, 56 (2012), 180-190.  doi: 10.1016/j.mcm.2011.12.041.

[14]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, 43, Springer Science & Business Media, 1999. doi: 10.1007/978-1-4612-1466-3.

[15]

X. Zeng, A stochastic differential reinsurance game, Journal of Applied Probability, 47, (2010), 335–349. doi: 10.1239/jap/1276784895.

[16]

G. Zhang, S. Cheng, Z. Li and M. Cao, A reinsurance and investment game between two insurers under the CEV model, Mathematical Problems in Engineering, 2020 (2020), 4696941, 12 pp. doi: 10.1155/2020/4696941.

[17]

M. Zhou and K.C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.  doi: 10.1016/j.mcm.2011.12.041.

[18]

J. Zhu, G. Guan and S. Li, Time-consistent non-zero-sum stochastic differential reinsurance and investment game under default and volatility risks, Journal of Computational and Applied Mathematics, 374, (2020). doi: 10.1016/j.cam.2020.112737.

show all references

References:
[1]

S. Asmussen and H. Albrecher, Ruin probabilities, 2$^{nd}$ edition World Scientific, Singapore, 2010. doi: 10.1142/9789814282536.

[2]

P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the cramér-lundberg model, Mathematical Finance, 15 (2005), 261-308.  doi: 10.1111/j.0960-1627.2005.00220.x.

[3]

N. Ba$\ddot{\text{u}}$erle, Approximation of optimal reinsurance and dividend payout policies, Mathematical Finance, 14 (2004), 99-113.  doi: 10.1111/j.0960-1627.2004.00183.x.

[4]

B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, in Transactions of the XVth International Congress of Actuaries, 2, (1957), 433–443.

[5]

D. LiX. Rong and H. Zhao, Stochastic differential game formulation on the reinsurance and investment problem, International Journal of Control, 88 (2015), 1861-1877.  doi: 10.1080/00207179.2015.1022797.

[6]

S. Luo, A stochastic differential game for quadratic-linear diffusion processes, Advances in Applied Probability, 48 (2016), 1161-1182.  doi: 10.1017/apr.2016.69.

[7]

S. Luo, M. Wang and W. Zhu, Stochastic differential reinsurance game in diffusion approximation models, Journal of Computational and Applied Mathematics, 386, (2021). doi: 10.1016/j.cam.2020.113252.

[8]

H. Meng, T. K. Siu and H. Yang, A note on optimal insurance risk control with multiple reinsurers, Journal of Computational and Applied Mathematics, 319, (2017), 38–42. doi: 10.1016/j.cam.2016.12.034.

[9]

H. Meng, T. K. Siu and H. Yang, Optimal insurance risk control with multiple reinsurers, Journal of Computational and Applied Mathematics, 306, (2016), 40–52. doi: 10.1016/j.cam.2016.04.005.

[10]

M. Taskar and X. Zeng, Optimal non-proportional reinsurance control and stochastic differential games, Insurance: Matematics and Economics, 48, (2011), 64–71. doi: 10.1016/j.insmatheco.2010.09.006.

[11]

N. Wang, N. Zhang and L. Qian, Reinsurance-investment game between two mean-variance insurers under model uncertainty, Journal of Computational and Applied Mathematics, 382, (2021). doi: 10.1016/j.cam.2020.113095.

[12]

W. Liu and Y.-J. Hu, Optimal proportional reinsurance and dividend payments with transaction costs and internal competition, Applied Mathematics-A Journal of Chinese Universities, 31 (2016), 89-102.  doi: 10.1007/s11766-016-3281-7.

[13]

J. Xu and M. Zhou, Optimal risk control and dividend distribution policies for a diffusion model with terminal value, Mathematical and Computer Modelling, 56 (2012), 180-190.  doi: 10.1016/j.mcm.2011.12.041.

[14]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, 43, Springer Science & Business Media, 1999. doi: 10.1007/978-1-4612-1466-3.

[15]

X. Zeng, A stochastic differential reinsurance game, Journal of Applied Probability, 47, (2010), 335–349. doi: 10.1239/jap/1276784895.

[16]

G. Zhang, S. Cheng, Z. Li and M. Cao, A reinsurance and investment game between two insurers under the CEV model, Mathematical Problems in Engineering, 2020 (2020), 4696941, 12 pp. doi: 10.1155/2020/4696941.

[17]

M. Zhou and K.C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.  doi: 10.1016/j.mcm.2011.12.041.

[18]

J. Zhu, G. Guan and S. Li, Time-consistent non-zero-sum stochastic differential reinsurance and investment game under default and volatility risks, Journal of Computational and Applied Mathematics, 374, (2020). doi: 10.1016/j.cam.2020.112737.

Figure 1.  Variance principles with correlation and dividends
Figure 2.  Variance principles with varying dividend payout
Figure 3.  Variance principles with varying correlation coefficient
Figure 4.  Expectation principles with correlation and dividend payout
Figure 5.  Expectation principles with varying dividend payout
Figure 6.  Expectation principles with varying correlation coefficient
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