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

. 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.

[1]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[2]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[3]  A. Tourin.  Splitting methods for Hamilton‐Jacobi equations , 2006 .

[4]  H. Ishii.  On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's , 1989 .

[5]  Hongjie Dong,et al.  On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients , 2006 .

[6]  P. Lions,et al.  Approximation numérique des équations Hamilton-Jacobi-Bellman , 1980 .

[7]  Maurizio Falcone,et al.  An approximation scheme for the optimal control of diffusion processes , 1995 .

[8]  Hamilton-Jacobi Equations,et al.  ON THE CONVERGENCE RATE OF APPROXIMATION SCHEMES FOR , 2022 .

[9]  P. Souganidis.  Approximation schemes for viscosity solutions of Hamilton-Jacobi equations , 1985 .

[10]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1990, 29th IEEE Conference on Decision and Control.

[11]  H. Ishii,et al.  Viscosity Solutions of a System of Nonlinear Second-Order Elliptic PDEs Arising in Switching Games , 2005 .

[12]  Espen R. Jakobsen,et al.  ON THE RATE OF CONVERGENCE OF APPROXIMATION SCHEMES FOR BELLMAN EQUATIONS ASSOCIATED WITH OPTIMAL STOPPING TIME PROBLEMS , 2003 .

[13]  Guy Barles,et al.  On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations , 2002 .

[14]  N. Krylov,et al.  Approximating Value Functions for Controlled Degenerate Diffusion Processes by Using Piece-Wise Constant Policies , 1999 .

[15]  N. V. Krylov.  The Rate of Convergence of Finite-Difference Approximations for Bellman Equations with Lipschitz Coefficients , 2004 .

[16]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[17]  J. Frédéric Bonnans,et al.  Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation , 2003, SIAM J. Numer. Anal..

[18]  Avner Friedman,et al.  Optimal stochastic switching and the Dirichlet problem for the Bellman equation , 1979 .

[19]  P. Lions,et al.  Two approximations of solutions of Hamilton-Jacobi equations , 1984 .

[20]  J. Frédéric Bonnans,et al.  A fast algorithm for the two dimensional HJB equation of stochastic control , 2004 .

[21]  N. Krylov.  On the rate of convergence of finite-difference approximations for Bellmans equations with variable coefficients , 2000 .

[22]  L. Evans,et al.  Optimal Switching for Ordinary Differential Equations , 1984 .

[23]  H. Ishii,et al.  Viscosity solutions for monotone systems of second-order elliptic PDES , 1991 .

[24]  Tyrone E. Duncan,et al.  Numerical Methods for Stochastic Control Problems in Continuous Time (Harold J. Kushner and Paul G. Dupuis) , 1994, SIAM Rev..

[25]  Nicolai V. Krylov,et al.  Rate of convergence of finite-difference approximations for degenerate linear parabolic equations with C1 and C2 coefficients , 2005 .

[26]  E. Jakobsen,et al.  Continuous Dependence Estimates for Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Equations , 2002 .

[27]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[28]  Hitoshi Ishii,et al.  The maximum principle for semicontinuous functions , 1990, Differential and Integral Equations.

[29]  Guy Barles,et al.  Error Bounds for Monotone Approximation Schemes for Hamilton-Jacobi-Bellman Equations , 2005, SIAM J. Numer. Anal..