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We investigate in this work a fully-discrete semi-Lagrangian approximation ofsecond order possibly degenerate Hamilton-Jacobi-Bellman (HJB) equations on abounded domain with oblique boundary conditions. These equations appearnaturally in the study of optimal control of diffusion processes with obliquereflection at the boundary of the domain.  The proposed scheme is shown to satisfy a consistency type property, it ismonotone and stable. Our main result is the convergence of the numericalsolution towards the unique viscosity solution of the HJB equation. Theconvergence result holds under the same asymptotic relation between the timeand space discretization steps as in the classical setting for semi-Lagrangianschemes. We present some numerical results that confirm the numericalconvergence of the scheme.

A semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with oblique boundary conditions