In this paper we deal with the problem of existence of a smooth solution ofthe Hamilton-Jacobi-Bellman-Isaacs (HJBI for short) system of equationsassociated with nonzero-sum stochastic differential games. We consider theproblem in unbounded domains either in the case of continuous generators or fordiscontinuous ones. In each case we show the existence of a smooth solution ofthe system. As a consequence, we show that the game has smooth Nash payoffswhich are given by means of the solution of the HJBI system and the stochasticprocess which governs the dynamic of the controlled system.
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