Approximation Schemes for Monotone Systems of Nonlinear Second Order Partial Differential Equations: Convergence Result and Error Estimate

We consider approximation schemes for monotone systems of fully nonlinear second order partial di erential equations. We rst prove a general convergence result for monotone, consistent and regular schemes. This result is a generalization to the well known framework of Barles-Souganidis, in the case of scalar nonlinear equation. Our second main result provides the convergence rate of approximation schemes for weakly coupled systems of Hamilton-Jacobi-Bellman equations. Examples including nite di erence schemes and Semi-Lagrangian schemes are discussed.

[1]  J. Frédéric Bonnans,et al.  Error estimates for stochastic differential games: the adverse stopping case , 2006 .

[2]  Guy Barles,et al.  Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations , 2007, Math. Comput..

[3]  Nizar Touzi,et al.  A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs , 2009, 0905.1863.

[4]  Fabio Camilli,et al.  A Finite Element Like Scheme for Integro-Partial Differential Hamilton-Jacobi-Bellman Equations , 2009, SIAM J. Numer. Anal..

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

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

[7]  M. K. Ghosh,et al.  Optimal control of switching diffusions with application to flexible manufacturing systems , 1993 .

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

[9]  R. González,et al.  On unbounded solutions of Bellman's equation associated with optimal switching control problems with state constraints , 1995 .

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

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

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

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

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

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

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

[17]  Espen R. Jakobsen,et al.  On Error Bounds for Approximation Schemes for Non-Convex Degenerate Elliptic Equations , 2004 .

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

[19]  Hitoshi Ishii,et al.  Perron's method for monotone systems of second-order elliptic partial differential equations , 1992, Differential and Integral Equations.

[20]  Panagiotis E. Souganidis,et al.  A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs , 2008 .

[21]  S. Lenhart,et al.  Viscosity Solutions for Weakly Coupled Systems of Hamilton-Jacobi Equations , 1991 .

[22]  R. Munos,et al.  Consistency of a simple multidimensional scheme for Hamilton–Jacobi–Bellman equations , 2005 .

[23]  M. Bladt,et al.  Pricing Derivatives Incorporating Structural Market Changes and in Time Correlation , 2008 .

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

[25]  Stefania Maroso,et al.  Error Estimates for a Stochastic Impulse Control Problem , 2007 .