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

Abstract

Abstract. We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order h 1/2 for certain types of finite-difference schemes are obtained.

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

“…One immediate consequence is Lipschitz continuity in the spatial variable of a viscosity solution, and with some additional reasoning also Hölder continuity in time. Another (recent) application concerns their relevance in Krylov's method of shaking the coefficients, which is used in numerical analysis of convex fully-nonlinear PDEs, see for example [34,10].…”

“…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.…”

On stochastic finite difference schemes

“…One can view them as (possibly degenerate) infinite systems of stochastic differential equations, whose components describe the time evolution of approximate values at the grid points of the solutions to SPDEs. Adapting the approach of [13] we view stochastic finite difference schemes, like in [7] and [3], as stochastic equations for random fields on the whole R d not only on grids.…”

“…One can view them as (possibly degenerate) infinite systems of stochastic differential equations, whose components describe the time evolution of approximate values at the grid points of the solutions to SPDEs. Adapting the approach of [13] we view stochastic finite difference schemes, like in [7] and [3], as stochastic equations for random fields on the whole R d not only on grids.Our aim is to investigate the rate of convergence in the supremum norm of the finite difference approximations. We show that under the stochastic parabolicity condition, if the coefficients and the data are sufficiently smooth, then the solutions to the finite difference schemes admit power series expansions in terms of h, the mesh-size of the grid.…”

Error estimates for finite difference-quadrature schemes for a class of nonlocal Bellman equations with variable diffusion

“…But the question of error estimate for numerical schemes, including finite difference schemes, is much more difficult and remained open until the recent works by Krylov [22,20,21] and Barles and Jakobsen [8,16,7].…”