Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

Abstract

We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $\sigma \in (0,2)$ since they involve fractional Laplace operators $(-\Delta {)}^{\sigma /2}$. They arise e.g. in control and game theory as dynamic programming equations -- HJB and Isaacs equation -- and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $\sigma$. The accuracy of previous approximations of fractional fully nonlinear equations depend on $\sigma$ and are worse when $\sigma$ is close to $2$. We show that the schemes are monotone, consistent, $L^{\infty }$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

23 pages, 5 figures, 2 tables