We consider the Hamilton-Jacobi-Bellman system
∂tu − ∆u = H(u,∇u) + f
for u ∈ RN, where the Hamiltonian H(u,∇u) satisfies a super-quadratic growth condition with respect to |∇u|. Such a nonlinear parabolic system corresponds to a stochastic differential game with N players. We obtain the existence of bounded weak solutions and prove regularity results in Sobolev spaces for the Dirichlet problem.