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We prove a priori estimates and regularity results for some quasilinear degenerate elliptic equations arising in optimal stochastic control problems. Our main results show that strong coerciveness of gradient terms forces bounded viscosity subsolutions to be globally Hölder continuous, and solutions to be locally Lipschitz continuous. We also give an existence result for the associated Dirichlet problem.doi:10.1090/s0002-9947-10-04807-5 fatcat:4h6hjypm25antejed5z3jnyvjm