# American Institute of Mathematical Sciences

September  2013, 18(7): 1889-1907. doi: 10.3934/dcdsb.2013.18.1889

## Optimal stochastic differential games with VaR constraints

 1 School of Insurance, Central University Of Finance and Economics, Beijing 100081, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  May 2011 Revised  January 2013 Published  May 2013

The nonlinear dynamic games between competing insurance companies are interesting and important problems because of the general practice of using re-insurance to reduce risks in the insurance industry. This problem becomes more complicated if a proper risk control is imposed on all the involving companies. In order to understand the dynamical properties, we consider the stochastic differential game between two insurance companies with risk constraints. The companies are allowed to purchase proportional reinsurance and invest their money into both risk free asset and risky (stock) asset. The competition between the two companies is formulated as a two player (zero-sum) stochastic differential game. One company chooses the optimal reinsurance and investment strategy in order to maximize the expected payoff, and the other one tries to minimize this value. For the purpose of risk management, the risk arising from the whole portfolio is constrained to some level. By the principle of dynamic programming, the problem is reduced to solving the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations for Nash equilibria. We derive the Nash equilibria explicitly and obtain closed form solutions to HJBI under different scenarios.
Citation: Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889
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