Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-18T06:00:19.520Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A NEW PARADIGM OF OPTIMAL REINSURANCE: A STOCHASTIC STACKELBERG DIFFERENTIAL GAME BETWEEN AN INSURER AND A REINSURER

Published online by Cambridge University Press:  26 March 2018

Lv Chen  and
Yang Shen
Lv Chen
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada School of Statistics, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China
Yang Shen*
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
*
E-mail: yangshen@yorku.ca, skyshen87@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper proposes a new continuous-time framework to analyze optimal reinsurance, in which an insurer and a reinsurer are two players of a stochastic Stackelberg differential game, i.e., a stochastic leader-follower differential game. This allows us to determine optimal reinsurance from joint interests of the insurer and the reinsurer, which is rarely considered in the continuous-time setting. In the Stackelberg game, the reinsurer moves first and the insurer does subsequently to achieve a Stackelberg equilibrium toward optimal reinsurance arrangement. Speaking more precisely, the reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal proportional reinsurance to purchase. Under utility maximization criteria, we study the game problem starting from the general setting with generic utilities and random coefficients to the special case with exponential utilities and constant coefficients. In the special case, we find that the reinsurer applies the variance premium principle to calculate the optimal reinsurance premium and the insurer's optimal ceding/retained proportion of insurance risk depends not only on the risk aversion of itself but also on that of the reinsurer.

Keywords

Stackelberg gameproportional reinsurancestochastic Hamilton-Jacobi-Bellman equationbackward stochastic differential equationvariance premium principle
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 2 , May 2018 , pp. 905 - 960
Copyright
Copyright © Astin Bulletin 2018 

References

Bensoussan, A., Chen, S. and Sethi, S.P. (2015) The maximum principle for global solutions of stochastic Stackelberg differential games. SIAM Journal on Control and Optimization, 53 (4), 1956–1981.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 (8), 2025–2037.Google Scholar
Browne, S. (2000) Stochastic differential portfolio games. Journal of Applied Probability, 37 (1), 126–147.CrossRefGoogle Scholar
Cai, J., Gerber, H.U. and Yang, H. (2006) Optimal dividends in an Ornstein-Uhlenbeck type model with credit and debit interest. North American Actuarial Journal, 10 (2), 94–108.CrossRefGoogle Scholar
Cai, J., Lemieux, C. and Liu, F. (2016) Optimal reinsurance from the perspectives of both an insurer and a reinsurer. ASTIN Bulletin, 46 (3), 815–849.CrossRefGoogle Scholar
Chen, L. and Shen, Y. (2017) Stochastic Stackelberg differential games under time-inconsistent mean-variance framework. Preprint.Google Scholar
Chen, L., Qian, L., Shen, Y. and Wang, W. (2016) Constrained investment-reinsurance optimization with regime switching under variance premium principle. Insurance: Mathematics and Economics, 71, 253–267.Google Scholar
Chen, P. and Yam, S.C.P. (2013) Optimal proportional reinsurance and investment with regime-switching for mean-variance insurers. Insurance: Mathematics and Economics, 53 (3), 871–883.Google Scholar
Chutani, A. and Sethi, S.P. (2012) Optimal advertising and pricing in a dynamic durable goods supply chain. Journal of Optimization Theory and Applications, 154 (2), 615–643.CrossRefGoogle Scholar
Cochrane, J.H. (2005) Asset Pricing (revised edition). Princeton, NJ: Princeton University Press.Google Scholar
Grandell, J. (1990) Aspects of Risk Theory. New York: Springer-Verlag.Google Scholar
He, X., Prasad, A. and Sethi, S.P. (2009) Cooperative advertising and pricing in a dynamic stochastic supply chain: Feedback Stackelberg strategies. Production and Operations Management, 18 (1), 78–94.Google Scholar
Hu, Y., Imkeller, P. and Müller, M. (2005) Utility maximization in incomplete markets. Annals of Applied Probability, 15 (3), 1691–1712.Google Scholar
Højgaard, B. and Taksar, M. (1998) Optimal proportional reinsurance policies for diffusion models. Scandinavian Actuarial Journal, 1998 (2), 166–180.Google Scholar
Jin, Z., Yin, G. and Wu, F. (2013) Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods. Insurance: Mathematics and Economics, 53 (3), 733–746.Google Scholar
Kobylanski, M. (2000) Backward stochastic differential equations and partial differential equations with quadratic growth. Annals of Probability, 28 (2), 558–602.Google Scholar
Kunita, H. (1981) Some extensions of Itô's formula. Séminaire de Probabilités de Strasbourg, 15, 118–141.Google Scholar
Li, D., Li, D. and Young, V.R. (2017) Optimality of excess-loss reinsurance under a mean-variance criterion. Insurance: Mathematics and Economics, 75, 82–89.Google Scholar
Lin, X., Zhang, C. and Siu, T.K. (2012) Stochastic differential portfolio games for an insurer in a jump-diffusion risk process. Mathematical Methods of Operations Research, 75 (1), 83–100.Google 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
Morlais, M.A. (2009) Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem. Finance and Stochastics, 13 (1), 121–150.Google Scholar
Øksendal, B., Sandal, L. and Ubøe, J. (2013) Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models. Journal of Economic Dynamics and Control, 37 (7), 1284–1299.Google Scholar
Øksendal, B. and Sulem, A. (2009) Maximum principles for optimal control of forward-backward stochastic differential equations with jumps. SIAM Journal on Control and Optimization, 48 (5), 2945–2976.Google Scholar
Peng, S. (1992) Stochastic Hamilton-Jacobi-Bellman equations. SIAM Journal on Control and Optimization, 30 (2), 284–304.Google Scholar
Promislow, S.D. and Young, V.R. (2005) Minimizing the probability of ruin when claims follow Brownian motion with drift. North American Actuarial Journal, 9 (3), 110–128.Google Scholar
Revuz, D. and Yor, M. (2005) Continuous Martingales and Brownian Motion (Corrected 3rd printing of 3rd ed.). Berlin/Heidelberg/New York: Springer-Verlag.Google Scholar
Shen, Y. and Wei, J. (2016) Optimal investment-consumption-insurance with random parameters. Scandinavian Actuarial Journal, 2016 (1), 37–62.Google Scholar
Shen, Y. and Zeng, Y. (2014) Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach. Insurance: Mathematics and Economics, 57, 1–12.Google Scholar
Shen, Y. and Zeng, Y. (2015) Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process. Insurance: Mathematics and Economics, 62, 118–137.Google Scholar
Shi, J., Wang, G. and Xiong, J. (2016) Leader-follower stochastic differential game with asymmetric information and applications. Automatica, 63, 60–73.Google Scholar
Taksar, M. and Zeng, X. (2011) Optimal non-proportional reinsurance control and stochastic differential games. Insurance: Mathematics and Economics, 48 (1), 64–71.Google Scholar
von Stackelberg, H. (1934) Marktform und Gleichgewicht. Wien und Berlin: Verlag von Julius Springer.Google Scholar
Yang, H. and Zhang, L. (2005) Optimal investment for insurer with jump-diffusion risk process. Insurance: Mathematics and Economics, 37 (3), 615–634.Google Scholar
Yong, J. (2002) A leader-follower stochastic linear quadratic differential game. SIAM Journal on Control and Optimization, 41 (4), 1015–1041.Google Scholar
Zeng, X. (2010) A stochastic differential reinsurance game. Journal of Applied Probability, 47 (2), 335–349.Google Scholar
Zeng, Y. and Li, Z. (2011) Optimal time-consistent investment and reinsurance policies for mean-variance insurers. Insurance: Mathematics and Economics, 49 (1), 145–154.Google Scholar
Zeng, Y., Li, Z. and Lai, Y. (2013) Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps. Insurance: Mathematics and Economics, 52 (3), 498–507.Google Scholar
Zhang, X. and Siu, T.K. (2009) Optimal investment and reinsurance of an insurer with model uncertainty. Insurance: Mathematics and Economics, 45 (1), 81–88.Google Scholar
Zhao, H., Shen, Y. and Zeng, Y. (2016) Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security. Journal of Mathematical Analysis and Applications, 437 (2), 1036–1057.CrossRefGoogle Scholar