Optimal stochastic differential games with VaR constraints

Abstract

The nonlinear dynamic games between competing insurance companies are interesting and important problemsbecause of the general practice of using re-insurance to reduce risks in the insurance industry. This problem becomes more complicatedif 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.