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