Nash points for nonzero-sum stochastic differential games with separate Hamiltonians

We study a nonzero-sum stochastic differential game under the assumptions that the control sets are multidimensional convex compact, the game has separate dynamic and running costs and the multifunctions representing the optimal feedbacks have convex values. To prove the existence of Nash equilibria we reduce to study a system of uniformly parabolic equations strongly coupled by multivalued applications. We obtain the existence of Nash points in two different cases: (i) R-valued process and general dynamic, (ii) multivalued process and affine dynamic.