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

翻译：在这项工作中,我们调查了一种完全分解的半拉格朗吉亚近似第二顺序,可能使汉密尔顿-贾科比-贝勒曼(HJB)等式在边界条件斜斜的封闭区方程式上出现。这些等式自然地出现在关于对扩散过程进行最佳控制的研究中,在域边界上斜反射。拟议的办法显示符合一致性类型属性,是单质的和稳定的。我们的主要结果是数字解决办法与HJB等式独特的粘度解决办法趋于一致。趋同的结果与半拉格朗江计划的传统环境一样,在时间和空间分解步骤之间有着同样的零星关系。我们提出了一些数字结果,证实了这个办法的数字趋同。