Precise Error Bounds for Numerical Approximations of Fractional HJB Equations

TL;DR

This paper establishes precise convergence rates for numerical schemes approximating fractional Hamilton-Jacobi-Bellman equations, accounting for solution regularity and improving upon previous results especially in degenerate cases.

Contribution

It provides optimal error estimates for both existing and new approximation schemes, capturing fractional order and solution regularity, with improvements over prior work in degenerate cases.

Findings

Optimal error estimates for strongly degenerate problems with Lipschitz solutions.

New convergence rates for weakly non-degenerate problems with fractional derivatives.

Improved rates exceeding 1/2 for certain fractional HJB equations.

Abstract

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.