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

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 nonconvex equations of order less than 2. In this paper they are also allowed to be degenerate and have nonsmooth 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 nonconvex 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 is, the rates differ when the nonlocal diffusion coefficient depends on $x$ and $t$, only on $x$, or on neither.