Appl Math Optim 56:37–66 (2007)
2007 Springer Science+Business Media, Inc.
The Rate of Convergence of Finite-Difference Approximations
for Parabolic Bellman Equations with Lipschitz
Coefﬁcients in Cylindrical Domains
and Nicolai V. Krylov
Department of Mathematics, University of Chicago,
5734 S. University Avenue, Chicago, IL 60637, USA
127 Vincent Hall, University of Minnesota,
Minneapolis, MN 55455, USA
Abstract. We consider degenerate parabolic and elliptic fully nonlinear Bellman
equations with Lipschitz coefﬁcients in domains. Error bounds of order h
sup norm for certain types of ﬁnite-difference schemes are obtained.
Key Words. Finite-difference approximations, Bellman equations, Fully non-
AMS Classiﬁcation. 65M15, 35J60, 93E20.
This article is a natural continuation of , where the rate of convergence of ﬁnite-
difference approximations in the sup norm for Bellman fully nonlinear parabolic equa-
tions in the whole space was investigated. Finite-difference equations on inﬁnite meshes
look not quite realistic from a practical point of view and here we deal with equa-
tions in bounded domains. As in  we give results for parabolic and elliptic equa-
tions. Although for an understanding of the proofs of our results the reader should
ﬁrst go through , for a broader audience we brieﬂy repeat part of the introduction
The work of the second author was partially supported by NSF Grant DMS-0140405.