Precise error bounds for numerical approximations of fractional HJB equations

Abstract

We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-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 nondegenerate problems where we show that solutions have bounded fractional derivatives of order $\sigma \in (1,2)$. 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 nondegenerate 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 $\mathcal{O}\big (h^{\frac{1}{2}}\big )$.