Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T12:56:53.541Z Has data issue: false hasContentIssue false
  • English
  • Français

A stochastic differential reinsurance game

Part of: Stochastic processes Stochastic systems and control Markov processes

Published online by Cambridge University Press:  14 July 2016

Xudong Zeng
Xudong Zeng*
Affiliation:
University of Missouri
*
∗Postal address: 217 Mathematical Sciences Building, University of Missouri, Columbia, Missouri, 65211, USA. Email address: zengxu@missouri.edu
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.

We study a stochastic differential game between two insurance companies who employ reinsurance to reduce the risk of exposure. Under the assumption that the companies have large insurance portfolios compared to any individual claim size, their surplus processes can be approximated by stochastic differential equations. We formulate competition between the two companies as a game with a single payoff function which depends on the surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize this expected payoff, while the other company simultaneously chooses a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of this stochastic differential game and solve it explicitly for the case of maximizing/minimizing the exit probability.

Keywords

Stochastic differential gameproportional reinsuranceCramer-Lundberg modelNash equilibriumstochastic control

MSC classification

Secondary: 93E20: Optimal stochastic control 60G40: Stopping times; optimal stopping problems; gambling theory 60J60: Diffusion processes
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 335 - 349
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Basar, T. and Olsder, G.J. (1982). Dynamic Noncooperative Game Theory. Academic Press, London.Google Scholar
[2] Browne, S. (2000). Stochastic differential portfolio games. J. Appl. Prob. 37, 126–147.CrossRefGoogle Scholar
[3] Elliott, R. J. (1976). The existence of value in stochastic differential games. SIAM J. Control Optimization 14, 85–94.CrossRefGoogle Scholar
[4] Elliott, R. J. and Davis, M. H. A. (1981). Optimal play in a stochatic differential game. SIAM J. Control Optimzation 19, 543–554.CrossRefGoogle Scholar
[5] Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, Berlin.CrossRefGoogle Scholar
[6] Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004). Loss Models, 2nd edn. John Wiley, Hoboken, NJ.Google Scholar
[7] Krylov, N. V. (1980). Controlled Diffusion Processes. Springer, New York.CrossRefGoogle Scholar
[8] Suijs, J., De Waegenaere, A. and Borm, P. (1998). Stochastic cooperative games in insurance. Insurance Math. Econom. 22, 209–228.CrossRefGoogle Scholar
[9] Taksar, M. I. (2000). Optimal risk and dividend distribution control models for an insurance company. Math. Meth. Operat. Res. 51, 1–42.CrossRefGoogle Scholar
[10] Taksar, M. I. and Markussen, C. (2003). Optimal dynamic reinsurance policies for large insurance portfolios. Finance Stoch. 7, 97–121.CrossRefGoogle Scholar
[11] Yeung, D. W. K. and Petrosyan, L. A. (2006). Cooperative Stochastic Differential Games. Springer, New York.Google Scholar
[12] Zeng, X. (2010). Optimal reinsurance with a rescuing procedure. Insurance Math. Econom. 46, 397–405.CrossRefGoogle Scholar