Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

Abstract

We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general – they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank–Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.