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Алгебра и анализ, 1997, том 9, выпуск 3, страницы 245–256 (Mi aa802)  

Эта публикация цитируется в 58 научных статьях (всего в 58 статьях)

On the rate of convergence of finite-difference approximations for Bellman's equations

N. V. Krylov

School of Mathematics, University of Minnesota, Minneapolis, MN

Аннотация: Some estimates are presented for the elliptic and parabolic Bellman equations with “constant” coefficients; these estimates are sharper than those announced earlier by the author in [8].

Ключевые слова: Finite-difference approximations, Bellman's equations, fully nonlinear equations.

Полный текст: PDF файл (561 kB)

Англоязычная версия:
St. Petersburg Mathematical Journal, 1998, 9:3, 639–650

Реферативные базы данных:

Поступила в редакцию: 26.12.1996
Язык публикации: английский

Образец цитирования: N. V. Krylov, “On the rate of convergence of finite-difference approximations for Bellman's equations”, Алгебра и анализ, 9:3 (1997), 245–256; St. Petersburg Math. J., 9:3 (1998), 639–650

Цитирование в формате AMSBIB
\RBibitem{Kry97}
\by N.~V.~Krylov
\paper On the rate of convergence of finite-difference
approximations for Bellman's equations
\jour Алгебра и анализ
\yr 1997
\vol 9
\issue 3
\pages 245--256
\mathnet{http://mi.mathnet.ru/aa802}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1466804}
\zmath{https://zbmath.org/?q=an:0902.65035}
\transl
\jour St. Petersburg Math. J.
\yr 1998
\vol 9
\issue 3
\pages 639--650


Образцы ссылок на эту страницу:
  • http://mi.mathnet.ru/aa802
  • http://mi.mathnet.ru/rus/aa/v9/i3/p245

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    Эта публикация цитируется в следующих статьяx:
    1. Krylov, NV, “On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients”, Probability Theory and Related Fields, 117:1 (2000), 1  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Krylov N.V., “Mean value theorems for stochastic integrals”, Annals of Probability, 29:1 (2001), 385–410  crossref  mathscinet  zmath  isi  scopus  scopus
    3. Barles, G, “On the convergence rate of approximation schemes for Hamilton–Jacobi–Bellman equations”, ESAIM-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique, 36:1 (2002), 33  crossref  mathscinet  zmath  isi  scopus  scopus
    4. Jakobsen, ER, “On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems”, Mathematical Models & Methods in Applied Sciences, 13:5 (2003), 613  crossref  mathscinet  zmath  isi  scopus  scopus
    5. Jakobsen, ER, “On error bounds for approximation schemes for non-convex degenerate elliptic equations”, BIT Numerical Mathematics, 44:2 (2004), 269  crossref  mathscinet  zmath  isi  scopus  scopus
    6. Hongjie Dong, N. V. Krylov, “On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients”, Алгебра и анализ, 17:2 (2005), 108–132  mathnet  mathscinet  zmath; St. Petersburg Math. J., 17:2 (2006), 295–313  crossref
    7. Cont, R, “A finite difference scheme for option pricing in jump diffusion and exponential Levy models”, SIAM Journal on Numerical Analysis, 43:4 (2005), 1596  crossref  mathscinet  zmath  isi  scopus  scopus
    8. Krylov, NV, “The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients”, Applied Mathematics and Optimization, 52:3 (2005), 365  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. Barles, G, “Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellma equations”, SIAM Journal on Numerical Analysis, 43:2 (2005), 540  crossref  mathscinet  zmath  isi  scopus  scopus
    10. Jakobsen E.R., Karlsen K.H., “Convergence rates for semi–discrete splitting approximations for degenerate parabolic equations with source terms”, BIT Numerical Mathematics, 45:1 (2005), 37–67  crossref  mathscinet  zmath  isi  scopus  scopus
    11. Jakobsen, ER, “On error bounds for monotone approximation schemes for multi-dimensional Isaacs equations”, Asymptotic Analysis, 49:3–4 (2006), 249  mathscinet  zmath  isi
    12. Bonnans, JF, “Error estimates for stochastic differential games: the adverse stopping case”, IMA Journal of Numerical Analysis, 26:1 (2006), 188  crossref  mathscinet  zmath  isi  scopus  scopus
    13. Zhang, JF, “Rate of convergence of finite difference approximations for degenerate ordinary differential equations”, Math. Comp., 75 (2006), 1755–1778  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    14. Barles, G, “Error bounds for monotone approximation schemes for parabolic Hamilton–Jacobi–Bellman equations”, Mathematics of Computation, 76:260 (2007), 1861  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    15. Dong, HJ, “The rate of convergence of finite-difference approximations for parabolic Bellman equations with Lipschitz coefficients in cylindrical domains”, Applied Mathematics and Optimization, 56:1 (2007), 37  crossref  mathscinet  zmath  isi  scopus  scopus
    16. Liang, J, “On the rate of convergence of the binomial tree scheme for American options”, Numerische Mathematik, 107:2 (2007), 333  crossref  mathscinet  zmath  isi  scopus  scopus
    17. Bonnans, F, “Error estimates for a stochastic impulse control problem”, Applied Mathematics and Optimization, 55:3 (2007), 327  crossref  mathscinet  zmath  isi  scopus  scopus
    18. Krylov N.V., “A priori estimates of smoothness of solutions to difference Bellman equations with linear and quasi–linear operators”, Mathematics of Computation, 76:258 (2007), 669–698  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    19. Song Q.S., Yin G., “Study of convergence rates of numerical methods for stochastic control problems”, Proceedings of the 46th IEEE Conference on Decision and Control, Vols 1–14, IEEE Conference on Decision and Control - Proceedings, 2007, 1162–1167  isi
    20. Luo, J, “On the Rate of Convergence of the Finite-Difference Approximations for Parabolic Bellman Equations with Constant Coefficients”, Applied Mathematics and Optimization, 58:3 (2008), 315  crossref  mathscinet  zmath  isi  scopus  scopus
    21. Jakobsen, ER, “Continuous dependence results for non-linear Neumann type boundary value problems”, Journal of Differential Equations, 245:9 (2008), 2368  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    22. Gyoengy I., Siska D., “On randomized stopping”, Bernoulli, 14:2 (2008), 352–361  crossref  mathscinet  zmath  isi  scopus  scopus
    23. Forcadel N., “An error estimate for a new scheme for mean curvature motion”, SIAM Journal on Numerical Analysis, 46:5 (2008), 2715–2741  crossref  mathscinet  zmath  isi  scopus  scopus
    24. Jakobsen E.R., Karlsen K.H., La Chioma C., “Error estimates for approximate solutions to Bellman equations associated with controlled jump–diffusions”, Numerische Mathematik, 110:2 (2008), 221–255  crossref  mathscinet  zmath  isi  scopus  scopus
    25. Biswas I.H., Jakobsen E.R., Karlsen K.H., “Error estimates for a class of finite difference–quadrature schemes for fully nonlinear degenerate parabolic integro–PDEs”, Journal of Hyperbolic Differential Equations, 5:1 (2008), 187–219  crossref  mathscinet  zmath  isi  scopus  scopus
    26. Caffarelli L.A., Souganidis P.E., “A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs”, Communications on Pure and Applied Mathematics, 61:1 (2008), 1–17  crossref  mathscinet  zmath  isi  scopus  scopus
    27. Song, QS, “RATES OF CONVERGENCE OF NUMERICAL METHODS FOR CONTROLLED REGIME-SWITCHING DIFFUSIONS WITH STOPPING TIMES IN THE COSTS”, SIAM Journal on Control and Optimization, 48:3 (2009), 1831  crossref  mathscinet  zmath  isi  scopus  scopus
    28. Hu, B, “Optimal convergence rate of the explicit finite difference scheme for American option valuation”, Journal of Computational and Applied Mathematics, 230:2 (2009), 583  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    29. Gyoengy I., Siska D., “On Finite–Difference Approximations for Normalized Bellman Equations”, Applied Mathematics and Optimization, 60:3 (2009), 297–339  crossref  mathscinet  zmath  isi  scopus  scopus
    30. Camilli F., Jakobsen E.R., “A Finite Element Like Scheme for Integro–Partial Differential Hamilton–Jacobi–Bellman Equations”, SIAM Journal on Numerical Analysis, 47:4 (2009), 2407–2431  crossref  mathscinet  zmath  isi  scopus  scopus
    31. Biswas I.H., Jakobsen E.R., Karlsen K.H., “Difference-Quadrature Schemes for Nonlinear Degenerate Parabolic Integro-PDE”, SIAM Journal on Numerical Analysis, 48:3 (2010), 1110–1135  crossref  mathscinet  zmath  isi  scopus  scopus
    32. Oberman A.M., “Convergence Rates for Difference Schemes for Polyhedral Nonlinear Parabolic Equations”, Journal of Computational Mathematics, 28:4 (2010), 474–488  mathscinet  zmath  isi
    33. Biswas I.H., Jakobsen E.R., Karlsen K.H., “Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes”, Applied Mathematics and Optimization, 62:1 (2010), 47–80  crossref  mathscinet  zmath  isi  scopus  scopus
    34. Song Q., Yin G., “Rates of Convergence of Markov Chain Approximation for Controlled Regime-switching Diffusions with Stopping Times”, 49th IEEE Conference on Decision and Control (CDC), 2010, 567–572  crossref  isi  scopus  scopus
    35. Arakelyan A., Barkhudaryan R., Poghosyan M., “An Error Estimate for the Finite Difference Scheme for One-Phase Obstacle Problem”, Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences, 46:3 (2011), 131–141  crossref  mathscinet  zmath  isi  scopus  scopus
    36. Fahim A., Touzi N., Warin X., “A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs”, Ann Appl Probab, 21:4 (2011), 1322–1364  crossref  mathscinet  zmath  isi  scopus
    37. Saleem M.Sh., Shashiashvili M., “Subsolutions That Are Close in the Uniform Norm Are Close in the Sobolev Norm as Well”, Appl. Math. Optim., 66:1 (2012), 1–25  crossref  mathscinet  zmath  isi  scopus
    38. Krylov N.V., “Interior Estimates for the First-Order Differences for Finite-Difference Approximations for Elliptic Bellman's Equations”, Appl. Math. Optim., 65:3 (2012), 349–370  crossref  mathscinet  zmath  isi  scopus  scopus
    39. Feng X., Glowinski R., Neilan M., “Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations”, SIAM Rev., 55:2 (2013), 205–267  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    40. Jensen M., Smears I., “On the Convergence of Finite Element Methods for Hamilton–Jacobi–Bellman Equations”, SIAM J. Numer. Anal., 51:1 (2013), 137–162  crossref  mathscinet  zmath  isi  scopus  scopus
    41. Krylov N.V., “Interior Estimates for Second-Order Differences of Solutions of Finite-Difference Elliptic Bellman's Equations”, Math. Comput., 82:283 (2013), 1463–1487  crossref  mathscinet  zmath  isi  scopus  scopus
    42. Krylov N.V., “On the Rate of Convergence of Difference Approximations For Uniformly Nondegenerate Elliptic Bellman's Equations”, Appl. Math. Optim., 69:3 (2014), 431–458  crossref  mathscinet  zmath  isi  scopus  scopus
    43. Bayraktar E., Fahim A., “A Stochastic Approximation For Fully Nonlinear Free Boundary Parabolic Problems”, Numer. Meth. Part Differ. Equ., 30:3 (2014), 902–929  crossref  mathscinet  zmath  isi  scopus  scopus
    44. Bokanowski O., Picarelli A., Zidani H., “Dynamic Programming and Error Estimates For Stochastic Control Problems With Maximum Cost”, Appl. Math. Optim., 71:1 (2015), 125–163  crossref  mathscinet  zmath  isi  scopus  scopus
    45. Turanova O., “Error Estimates For Approximations of Nonhomogeneous Nonlinear Uniformly Elliptic Equations”, Calc. Var. Partial Differ. Equ., 54:3 (2015), 2939–2983  crossref  mathscinet  zmath  isi  scopus  scopus
    46. Assellaou M., Bokanowski O., Zidani H., “Error Estimates For Second Order Hamilton–Jacobi–Bellman Equations. Approximation of Probabilistic Reachable Sets”, Discret. Contin. Dyn. Syst., 35:9, SI (2015), 3933–3964  crossref  mathscinet  zmath  isi  scopus  scopus
    47. Krylov N.V., “on the Rate of Convergence of Finite-Difference Approximations For Elliptic Isaacs Equations in Smooth Domains”, Commun. Partial Differ. Equ., 40:8 (2015), 1393–1407  crossref  mathscinet  zmath  isi  scopus  scopus
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    49. Guo W., Zhang J., Zhuo J., “a Monotone Scheme For High-Dimensional Fully Nonlinear PDEs”, Ann. Appl. Probab., 25:3 (2015), 1540–1580  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    50. Reisinger C., Forsyth P.A., “Piecewise constant policy approximations to Hamilton–Jacobi–Bellman equations”, Appl. Numer. Math., 103 (2016), 27–47  crossref  mathscinet  zmath  isi  scopus
    51. Neilan M., Salgado A.J., Zhang W., “Numerical Analysis of Strongly Nonlinear PDEs”, Acta Numer., 26 (2017), 137–303  crossref  mathscinet  zmath  isi  scopus
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    53. Bokanowski O., Picarelli A., Reisinger Ch., “High-Order Filtered Schemes For Time-Dependent Second Order Hjb Equations”, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 52:1 (2018), 69–97  crossref  isi
    54. Li Ya. Zhang Zh. Hu B., “Convergence Rate of An Explicit Finite Difference Scheme For a Credit Rating Migration Problem”, SIAM J. Numer. Anal., 56:4 (2018), 2430–2460  crossref  mathscinet  zmath  isi  scopus
    55. Arakelyan A., Barkhudaryan R., Shahgholian H., Salehi M., “Numerical Treatment to a Non-Local Parabolic Free Boundary Problem Arising in Financial Bubbles”, Bull. Iran Math. Soc., 45:1 (2019), 59–73  crossref  mathscinet  zmath  isi  scopus
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