Optimal stochastic differential  games  with VaR constraints

Jingzhen Liu; Ka-Fai Cedric Yiu

doi:10.3934/dcdsb.2013.18.1889

Article Contents

2013, Volume 18, Issue 7: 1889-1907. Doi: 10.3934/dcdsb.2013.18.1889

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Optimal stochastic differential games with VaR constraints

Jingzhen Liu^1, and
Ka-Fai Cedric Yiu^2,

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

Received: April 30, 2011

Revised: December 31, 2012

Published: August 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 93E20, 91A25; Secondary: 49l20.

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