On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman Equations

Abstract

Abstract. Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-JacobiBellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.Mathematics Subject Classification. 65N06, 65N15, 41A25, 49L20, 49L25.

On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients

“…This theorem is close to Theorem 3.5 in [1], where in a particular elliptic situation the error bound of order of accuracy h 1/2 was obtained on the account of a special approximation. The authors of [1] and [2] did a very good job of surveying the literature related to finite-difference approximations for the Hamilton-Jacobi and Bellman equations; instead of copying their comments, we allow ourselves to refer the interested reader to [1] and [2] for that information.…”

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

“…This is a difficult problem that remained open for a long time before the works of Krylov [32,33,34] and Barles & Jakobsen [8,9,10]. The methods developed in these works involve the use of carefully chosen smooth approximations of the viscosity solution of the underlying equation.…”

The Rate of Convergence of Finite-Difference Approximations for Bellman Equations with Lipschitz Coefficients

“…We refer to [2] and [3] and the references therein for discussion of what is achieved in this direction.…”