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A stochastic differential reinsurance game

Part of: Markov processes Stochastic systems and control Stochastic 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
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Abstract

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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 

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