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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
Posted:
January 8, 2007
MathSciNet review:
2291833
Full-text PDF Free Access
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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.
- 1.
J.
Frédéric Bonnans, Élisabeth
Ottenwaelter, and Housnaa
Zidani, A fast algorithm for the two dimensional HJB equation of
stochastic control, M2AN Math. Model. Numer. Anal. 38
(2004), no. 4, 723–735. MR 2087732
(2005e:93165), http://dx.doi.org/10.1051/m2an:2004034
- 2.
Guy
Barles and Espen
Robstad Jakobsen, On the convergence rate of approximation schemes
for Hamilton-Jacobi-Bellman equations, M2AN Math. Model. Numer. Anal.
36 (2002), no. 1, 33–54. MR 1916291
(2003h:65142), http://dx.doi.org/10.1051/m2an:2002002
- 3.
Guy
Barles and Espen
R. Jakobsen, Error bounds for monotone approximation schemes for
Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal.
43 (2005), no. 2, 540–558 (electronic). MR 2177879
(2007h:65116), http://dx.doi.org/10.1137/S003614290343815X
- 4.
Barles, G., and Jakobsen, E.R., Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations, preprint.
- 5.
Hongjie
Dong and N.
V. Krylov, On the rate of convergence of
finite-difference approximations for Bellman equations with constant
coefficients, Algebra i Analiz 17 (2005),
no. 2, 108–132; English transl., St.
Petersburg Math. J. 17 (2006), no. 2, 295–313. MR 2159586
(2006f:49050), http://dx.doi.org/10.1090/S1061-0022-06-00905-8
- 6.
Hongjie
Dong and Nicolai
V. Krylov, Rate of convergence of finite-difference approximations
for degenerate linear parabolic equations with 𝐶¹ and
𝐶² coefficients, Electron. J. Differential Equations
(2005), No. 102, 25. 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.
Espen
Robstad Jakobsen, On the rate of convergence of approximation
schemes for Bellman equations associated with optimal stopping time
problems, Math. Models Methods Appl. Sci. 13 (2003),
no. 5, 613–644. MR 1978929
(2004g:49054), http://dx.doi.org/10.1142/S0218202503002660
- 9.
N.
V. Krylov, Controlled diffusion processes, Applications of
Mathematics, vol. 14, Springer-Verlag, New York, 1980. Translated from
the Russian by A. B. Aries. MR 601776
(82a:60062)
- 10.
N.
V. Krylov, On the rate of convergence of finite-difference
approximations for Bellman’s equations, Algebra i Analiz
9 (1997), no. 3, 245–256; English transl., St.
Petersburg Math. J. 9 (1998), no. 3, 639–650.
MR
1466804 (98h:49033)
- 11.
N.
V. Krylov, Approximating value functions for controlled degenerate
diffusion processes by using piece-wise constant policies, Electron.
J. Probab. 4 (1999), no. 2, 19. MR 1668597
(2000b:49056), http://dx.doi.org/10.1214/EJP.v4-39
- 12.
N.
V. Krylov, On the rate of convergence of finite-difference
approximations for Bellman’s equations with variable
coefficients, Probab. Theory Related Fields 117
(2000), no. 1, 1–16. MR 1759507
(2001j:65134), http://dx.doi.org/10.1007/s004400050264
- 13.
Nicolai
V. Krylov, The rate of convergence of finite-difference
approximations for Bellman equations with Lipschitz coefficients,
Appl. Math. Optim. 52 (2005), no. 3, 365–399.
MR
2174020 (2006k:65219), http://dx.doi.org/10.1007/s00245-005-0832-3
- 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
 and  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)
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Additional Information
N. V. Krylov
Affiliation:
Department of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, Minnesota 55455
Email:
krylov@math.umn.edu
DOI:
http://dx.doi.org/10.1090/S0025-5718-07-01953-9
PII:
S 0025-5718(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
Posted:
January 8, 2007
Additional Notes:
The work was partially supported by NSF Grant DMS-0140405
Article copyright:
© Copyright 2007 American Mathematical Society
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