We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton-Jacobi-Bellman (HJB) equations. We consider diffusion corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations which are (formally) fractional order and 2nd order methods. It is well-known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) Strongly degenerate problems with Lipschitz solutions, and (ii) weakly non-degenerate problems where we show that solutions have bounded fractional derivatives of order between 1 and 2. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly non-degenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than 1/2.

翻译：我们证明了分数阶与非局部Hamilton-Jacobi-Bellman（HJB）方程单调逼近格式的精确收敛速率。我们考虑了文献中已有的扩散修正差分-求积格式，以及基于离散拉普拉斯算子幂次的新型逼近方法——这些格式（形式上）属于分数阶方法和二阶方法。数值分析中众所周知，收敛速率依赖于解的正则性，此处我们考虑了不同解正则性的情形：（i）具有Lipschitz解的强退化问题，以及（ii）弱非退化问题——我们证明此类问题的解具有介于1到2之间的有界分数阶导数。我们的主要成果是最优误差估计，其收敛速率精确刻画了格式的分数阶次与解的分数阶正则性。对于强退化方程，这些速率改进了早期结果；对于阶数大于1的弱非退化问题，所得结论为全新发现。在此我们证明，相较于强退化情形，弱非退化问题具有更优的收敛速率，且始终优于1/2。