On the rate of convergence for monotone numerical schemes for nonlocal Isaacs' equations

TL;DR

This paper establishes new error estimates for monotone numerical schemes solving nonlocal Isaacs equations, which are fully nonlinear, possibly degenerate, and depend on fractional order, with rates varying based on coefficient dependencies.

Contribution

It provides the first a priori error estimates for nonlocal Isaacs equations, including degenerate non-convex cases, with explicit dependence on fractional order and coefficient variability.

Findings

First error estimates for nonlocal Isaacs equations

Rates differ based on diffusion coefficient dependencies

Addresses degenerate, non-smooth solutions in viscosity setting

Abstract

We study monotone numerical schemes for nonlocal Isaacs equations, the dynamic programming equations of stochastic differential games with jump-diffusion state processes. These equations are fully-nonlinear non-convex equations of order less than $2$. In this paper they are also allowed to be degenerate and have non-smooth solutions. The main contribution is a series of new a priori error estimates: The first results for nonlocal Isaacs equations, the first general results for degenerate non-convex equations of order greater than $1$, and the first results in the viscosity solution setting giving the precise dependence on the fractional order of the equation. We also observe a new phenomena, that the rates differ when the nonlocal diffusion coefficient depend on $x$ and $t$, only on $x$, or on neither.