On error bounds for monotone approximation schemes for multi-dimensional Isaacs equations

摘要

Recently, Krylov, Barles, and Jakobsen developed the theory for estimating errors of monotone approximation schemes for the Bellman equation (a convex Isaacs equation). In this paper we consider an extension of this theory to a class of non-convex multidimensional Isaacs equations. This is the first result of this kind for non-convex multidimensional fully non-linear problems. To get the error bound, a key intermediate step is to introduce a penalization approximation. We conclude by (ⅰ) providing new error bounds for penalization approximations extending earlier results by, e.g., Bensoussan and Lions, and (ⅱ) obtaining error bounds for approximation schemes for the penalization equation using very precise a priori bounds and a slight generalization of the recent theory of Krylov, Barles, and Jakobsen.