Systems of Bellman equations to stochastic differential games with discount control

We consider two dimensional diagonal elliptic systems Δu+ au = H(x, u, ∇u) which arise from stochastic differential games with discount control. The Hamiltonians H have quadratic growth in ∇u and a special structure which has not yet been covered by regularity theory. Without smallness condition on H, the existence of a regular solution is established.