Appl Math Optim 52:365–399 (2005)
2005 Springer Science+Business Media, Inc.
The Rate of Convergence of Finite-Difference Approximations
for Bellman Equations with Lipschitz Coefﬁcients
Nicolai V. Krylov
127 Vincent Hall, University of Minnesota,
Minneapolis, MN 55455, USA
Abstract. We consider parabolic Bellman equations with Lipschitz coefﬁcients.
Error bounds of order h
for certain types of ﬁnite-difference schemes are obtained.
Key Words. Finite-difference approximations, Bellman equations, Fully non-
AMS Classiﬁcation. 65M15, 35J60, 93E20.
Bellman equations arise in many areas of mathematics, say in control theory, differential
geometry, and mathematical ﬁnance, to name a few. These equations typically are fully
nonlinear second-order degenerate elliptic or parabolic equations. In the particular case
of complete degeneration they become Hamilton–Jacobi ﬁrst-order equations.
Quite naturally, the problem of ﬁnding numerical methods of approximating solu-
tions to Bellman equations arises. First methods dating back some 30 years ago were
based on the fact that the solutions are the value functions in certain problems for con-
trolled diffusion processes, that can be approximated by controlled Markov chains. An
account of the results obtained in this direction can be found in , , and . These
results are based, in part, on the theory of weak convergence of probability measures
and, in part, on a remarkable result of .
Another approach is based on the notion of viscosity solution, which allows one
to avoid using probability theory. We refer to  and  and the references therein for
discussion of what is achieved in this direction.
This work was partially supported by NSF Grant DMS-0140405.