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：本文证明了分数阶及非局部Hamilton-Jacobi-Bellman (HJB)方程单调逼近格式的精确收敛速度。我们考虑了文献中基于扩散修正的差分求积格式，以及基于离散拉普拉斯算子幂次的新型逼近方法，这些方法在形式上是分数阶和二阶方法。数值分析中公认收敛速度依赖于解的正则性，本文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阶之间的有界分数阶导数。我们的主要结果是最优误差估计，其收敛速度精确刻画了格式的分数阶次和解的分数正则性。对于强退化方程，这些速度改进了先前结果；对于阶数大于1的弱非退化问题，本文结果属于首次提出。我们证明了相较于强退化情形有所提升的收敛速度，这些速度始终优于1/2。