Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T03:27:06.375Z Has data issue: false hasContentIssue false
  • English
  • Français

STOCHASTIC DIFFERENTIAL GAMES BETWEEN TWO INSURERS WITH GENERALIZED MEAN-VARIANCE PREMIUM PRINCIPLE

Published online by Cambridge University Press:  19 January 2018

Shumin Chen ,
Hailiang Yang  and
Yan Zeng
Shumin Chen
Affiliation:
School of Management, Guangdong University of Technology, Guangzhou, China, E-Mail: [email protected]
Hailiang Yang
Affiliation:
Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, China, E-Mail: [email protected]
Yan Zeng*
Affiliation:
Lingnan (University) College, Sun Yat-sen University, Guangzhou, China
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a stochastic differential game problem between two insurers, who invest in a financial market and adopt reinsurance to manage their claim risks. Supposing that their reinsurance premium rates are calculated according to the generalized mean-variance principle, we consider the competition between the two insurers as a non-zero sum stochastic differential game. Using dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman equations and show the existence of equilibrium strategies. For an exponential utility maximizing game and a probability maximizing game, we obtain semi-explicit solutions for the equilibrium strategies and the equilibrium value functions, respectively. Finally, we provide some detailed comparative-static analyses on the equilibrium strategies and illustrate some economic insights.

Keywords

Reinsurancegeneralized mean-variance premium principlenon-zero sum gameequilibrium strategyHamilton-Jacobi-Bellman equation
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 1 , January 2018 , pp. 413 - 434
Copyright
Copyright © Astin Bulletin 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bai, L., Cai, J. and Zhou, M. (2013) Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting. Insurance: Mathematics and Economics, 53, 664–670.Google Scholar
Bensoussan, A., Siu, C.C., Yam, S.C.P. and Yang, H. (2014). A class of non-zero-sum stochastic differential investment and reinsurance games. Automatica, 50, 2025–2037.CrossRefGoogle Scholar
Bi, J., Liang, Z. and Xu, F. (2016). Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence. Insurance: Mathematics and Economics, 70, 245–258.Google Scholar
Browne, S. (1995) Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Mathematics of Operations Research, 20, 937–958.Google Scholar
Browne, S. (2000) Stochastic differential portfolio games. Journal of Applied Probability, 37, 126–147.Google Scholar
Bühlmann, H. (1970). Mathematical Models in Risk Theory. New York: Springer.Google Scholar
Chi, Y. (2012) Optimal reinsurance under variance related premium principles. Insurance: Mathematics and Economics, 51, 310–321.Google Scholar
Chiu, M.C. and Wong, H.Y. (2014) Mean-variance asset-liability management with asset correlation risk and insurance liabilities. Insurance: Mathematics and Economics, 59, 300–310.Google Scholar
Espinosa, G.E. and Touzi, N. (2015) Optimal investment under relative performance concerns. Mathematical Finance, 25, 221–257.CrossRefGoogle Scholar
Gerber, H. (1970) Mathematical Methods in Risk Theory. Berlin: Springer.Google Scholar
Golubin, A.Y. (2008) Optimal insurance and reinsurance policies in the risk process. ASTIN Bulletin, 38, 383–397.CrossRefGoogle Scholar
Grandell, J. (1991) Aspects of Risk Theory. New York: Springer.CrossRefGoogle Scholar
Liang, Z. and Yuen, K.C. (2016). Optimal dynamic reinsurance with dependent risks: variance premium principle. Scandinavian Actuarial Journal, 1, 18–36.CrossRefGoogle Scholar
Meng, H., Li, S. and Jin, Z. (2015) A reinsurance game between two insurance companies with nonlinear risk processes. Insurance: Mathematics and Economics, 62, 91–97.Google Scholar
Pun, C.S., Siu, C.C. and Wong, H.Y. (2016) Non-zero-sum reinsurance games subject to ambiguous correlations. Operations Research Letters, 44, 578–586.Google Scholar
Pun, C.S. and Wong, H.Y. (2016) Robust non-zero-sum stochastic differential reinsurance game. Insurance: Mathematics and Economics, 68, 169–177.Google Scholar
Schmidli, H. (2008) Stochastic Control in Insurance (Probability and Its Applications). Berlin: Springer.Google Scholar
Siu, C.C., Yam, S.C.P., Yang, H. and Zhao, H. (2016). A class of nonzero-sum investment and reinsurance games subject to systematic risks. Scandinavian Actuarial Journal, 2016, 1–38.Google Scholar
Taksar, M. and Zeng, X. (2011) Optimal non-proportional reinsurance control and stochastic differential games. Insurance: Mathematics and Economics, 48, 64–71.Google Scholar
Yang, H. and Zhang, L. (2005) Optimal investment for insurer with jump-diffusion risk process. Insurance: Mathematics and Economics, 37, 615–634.Google Scholar
Yong, J. and Zhou, X.Y. (1999) Stochastic controls: Hamiltonian systems and HJB equations. New York: Springer.Google Scholar
Zeng, X. (2010) A stochastic differential reinsurance game. Journal of Applied Probability, 47, 335–349.Google Scholar
Zeng, X. and Luo, S. (2013) Stochastic Pareto-optimal reinsurance policies. Insurance: Mathematics and Economics, 53, 671–677.Google Scholar
Zeng, Y., Li, D. and Gu, A. (2016) Robust equilibrium reinsurance-investment strategy for a meanCvariance insurer in a model with jumps. Insurance: Mathematics and Economics, 66, 138–152.Google Scholar
Zhang, X., Meng, H. and Zeng, Y. (2016) Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling. Insurance: Mathematics and Economics, 53, 671–677.Google Scholar