On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients
type: Item
instance of: scholarly article
title: On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients
zbMATH Open document ID: 0971.65081
publication date: 2001-03-12T00:00:00Z
review text: Finite difference schemes to generate approximate viscosity or probabilistic solutions for degenerate Bellman's equation are considered, estimates of their convergence rates being deduced. For fixed real \(T>0\) the studied problem reads as \[ \begin{aligned} F\left(\frac{\partial u}{\partial t}, {\mathbf D }_2 u , \nabla u , u , t, x \right) = 0 &\quad \text{in } (0,T) \times \mathbb{R}^d \\ u(T,x) = g(x)&\quad \text{in } \mathbb{R}^d \end{aligned} \tag{1} \] where the sought scalar function \(u\) ought to be defined on the cylinder \({\mathcal C}=(0,T) \times \mathbb{R}^d\), being \(\mathbb{R}^d\) the \(d\)-dimensional Euclidean space, and \({\mathbf D }_2 u = ( \partial ^2 u /\partial x_\imath \partial x_\jmath)_{\imath \jmath } \). In (1), \(F\) denotes -- for real \(t\in (0,T)\) and \(x\in\mathbb{R}^d\), for given scalar functions \(v\), \(\omega\) and \(d\)-dimensional square-matrix and vector functions \( \mathbf y \) and \( \mathbf z \), respectively, all of them having \({\mathcal C}\) as domain, -- the functional defined by \[ F( v,{\mathbf y } , {\mathbf z }, \omega, t, x):= \sup _{\alpha } \{v + \sum_{\imath , \jmath =1}^{d} { a }^{\imath \jmath}(\alpha,t,x){ y }_{\imath \jmath} + \sum_{\imath =1 }^{d}{ b }^{\imath }(\alpha,t,x){ z }_{\imath} - c^{\alpha}(t,x)\omega + f^{\alpha}(t,x)\}. \] Here the supremum must be taken for \(\alpha\) running on the set of all admissible controls -- a separable metric space denoted by \(A\) -- and the coefficients \({\mathbf a }= (a^{\imath \jmath}),{\mathbf b }=({ b }^{\imath }),c\), and the forcing term \(f\) are all given on \(A\times{\mathcal C}\). This kind of differential system is related to dynamic programming equations for value functions in control problems of diffusion processes. Convergence proofs for numerical solutions to this problem are obtained mainly with two methods. The first one is due to \textit{H. J. Kushner} and \textit{P. G. Dupuis} [Numerical methods for stochastic control problems in continuou…
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 65M06
zbMATH DE Number: 1484726
zbMATH Keywords: Bellman's equation
zbMATH Keywords: dynamic programming
zbMATH Keywords: control
zbMATH Keywords: diffusion process
zbMATH Keywords: finite-differences
zbMATH Keywords: viscosity solution
zbMATH Keywords: value function
zbMATH Keywords: convergence
zbMATH Keywords: Markov chains
author: N. V. Krylov
MaRDI profile type: Publication
DOI: 10.1007/S004409900044
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
Recommended article: On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients
Recommended article: Q4396657
Recommended article: On the rate of convergence of the finite-difference approximations for parabolic Bellman equations with constant coefficients
Recommended article: The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients
Recommended article: On the rate of convergence of difference approximations for uniformly nondegenerate elliptic Bellman's equations
Recommended article: The rate of convergence of finite-difference approximations for parabolic bellman equations with Lipschitz coefficients in cylindrical domains
Recommended article: On finite-difference approximations for normalized Bellman equations
Recommended article: On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman Equations
Recommended article: ON THE RATE OF CONVERGENCE OF APPROXIMATION SCHEMES FOR BELLMAN EQUATIONS ASSOCIATED WITH OPTIMAL STOPPING TIME PROBLEMS
description: scientific article; zbMATH DE number 1484726