A class of non-zero-sum stochastic differential investment and reinsurance games |
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Authors: | Alain Bensoussan Chi Chung Siu Sheung Chi Phillip Yam Hailiang Yang |
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Affiliation: | 1. International Center for Decision and Risk Analysis, School of Management, The University of Texas at Dallas, Richardson, TX, USA;2. Department of Systems Engineering and Engineering Management, College of Science and Engineering, City University of Hong Kong, Hong Kong, China;3. School of Business, The University of Technology, Sydney, NSW, Australia;4. Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China;5. Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China |
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Abstract: | In this article, we provide a systematic study on the non-zero-sum stochastic differential investment and reinsurance game between two insurance companies. Each insurance company’s surplus process consists of a proportional reinsurance protection and an investment in risky and risk-free assets. Each insurance company is assumed to maximize his utility of the difference between his terminal surplus and that of his competitor. The surplus process of each insurance company is modeled by a mixed regime-switching Cramer–Lundberg diffusion approximation process, i.e. the coefficients of the diffusion risk processes are modulated by a continuous-time Markov chain and an independent market-index process. Correlation between the two surplus processes, independent of the risky asset process, is allowed. Despite the complex structure, we manage to solve the resulting non-zero sum game problem by applying the dynamic programming principle. The Nash equilibrium, the optimal reinsurance/investment, and the resulting value processes of the insurance companies are obtained in closed forms, together with sound economic interpretations, for the case of an exponential utility function. |
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Keywords: | Hamiltonian&ndash Jacobi&ndash Bellman equation Non-zero-sum stochastic differential game Equilibrium investment Equilibrium proportional reinsurance Regime switching Relative performance Cramer&ndash Lundberg model Nash equilibrium Stochastic control |
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