Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A priori estimates of smoothness of solutions to difference Bellman equations with linear and quasi-linear operators


Author: N. V. Krylov
Journal: Math. Comp. 76 (2007), 669-698
MSC (2000): Primary 65M15, 35J60, 93E20
DOI: https://doi.org/10.1090/S0025-5718-07-01953-9
Published electronically: January 8, 2007
MathSciNet review: 2291833
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A priori estimates for finite-difference approximations for the first and second-order derivatives are obtained for solutions of parabolic equations described in the title.


References [Enhancements On Off] (What's this?)

  • 1. Bonnans, J. F., Ottenwaelter, E., and Zidani, H., A fast algorithm for the two dimensional HJB equation of stochastic control, M2AN Math. Model. Numer. Anal., Vol. 38 (2004), No. 4, 723-735. MR 2087732 (2005e:93165)
  • 2. Barles, G., and Jakobsen, E.R., On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, Mathematical Modelling and Numerical Analysis, ESAIM, M2AM, Vol. 36 (2002), No. 1, 33-54. MR 1916291 (2003h:65142)
  • 3. Barles, G., and Jakobsen, E.R., Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal., Vol. 43 (2005), No. 2, 540-558. MR 2177879
  • 4. Barles, G., and Jakobsen, E.R., Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations, preprint.
  • 5. Dong, Hongjie and Krylov, N.V., On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients, Algebra i Analiz (St. Petersburg Math. J.), Vol. 17 (2005), No. 2, 108-132. MR 2159586 (2006f:49050)
  • 6. Dong, Hongjie and Krylov, N.V., On the rate of convergence of finite-difference approximations for degenerate linear parabolic equations with $ C^{1}$ and $ C^{2}$ coefficients, Electron. J. Diff. Eqns., Vol. 2005 (2005), No. 102, pp. 1-25. http://ejde.math.txstate.edu MR 2162263 (2006i:35008)
  • 7. Dong, Hongjie and Krylov, N.V., On the rate of convergence of finite-difference approximations for parabolic Bellman equations with Lipschitz coefficients in cylindrical domains, submitted to Applied Math. and Optimization.
  • 8. Jakobsen, E. R., On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems, Math. Models Methods Appl. Sci., Vol. 13 (2003), No. 5, 613-644. MR 1978929 (2004g:49054)
  • 9. Krylov, N.V., ``Controlled diffusion processes'', Nauka, Moscow, 1977 in Russian; English translation Springer, 1980. MR 601776 (82a:60062)
  • 10. Krylov, N.V., On the rate of convergence of finite-difference approximations for Bellman's equations, Algebra i Analiz, St. Petersburg Math. J., Vol. 9 (1997), No. 3, 245-256. MR 1466804 (98h:49033)
  • 11. Krylov, N.V., Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies, Electronic Journal of Probability, Vol. 4 (1999), paper no. 2, 1-19, http://www.math.washington.edu/~ejpecp/EjpVol4/ paper2.abs.html. MR 1668597 (2000b:49056)
  • 12. Krylov, N.V., On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probab. Theory Relat. Fields, Vol. 117 (2000), No. 1, 1-16. MR 1759507 (2001j:65134)
  • 13. Krylov, N.V., The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients, Applied Math. and Optimization, Vol. 52 (2005), No. 3, 365-399. MR 2174020 (2006k:65219)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M15, 35J60, 93E20

Retrieve articles in all journals with MSC (2000): 65M15, 35J60, 93E20


Additional Information

N. V. Krylov
Affiliation: Department of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, Minnesota 55455
Email: krylov@math.umn.edu

DOI: https://doi.org/10.1090/S0025-5718-07-01953-9
Keywords: Finite-difference approximations, Bellman equations, fully nonlinear equations.
Received by editor(s): November 13, 2005
Received by editor(s) in revised form: May 14, 2006
Published electronically: January 8, 2007
Additional Notes: The work was partially supported by NSF Grant DMS-0140405
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society