Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 

 

Rate of convergence of discrete approximate solutions of stochastic differential equations in a Hilbert space


Author: G. Shevchenko
Translated by: The author
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 69 (2003).
Journal: Theor. Probability and Math. Statist. 69 (2004), 187-199
MSC (2000): Primary 60H35; Secondary 60H10, 60H20, 65C30
DOI: https://doi.org/10.1090/S0094-9000-05-00625-3
Published electronically: February 9, 2005
MathSciNet review: 2110916
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider discrete-time approximations for stochastic differential equations in a Hilbert space. The rate of convergence of approximations is established for equations with Lipschitz continuous coefficients and for semilinear evolution type equations with an unbounded drift. As an auxiliary result, the rate of convergence of approximations is obtained for Itô-Volterra equations in a Hilbert space.


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

  • 1. Dorel Barbu and Viorel Radu, Approximations to mild solutions of stochastic semilinear equations, Novi Sad J. Math. 30 (2000), no. 1, 183–190. MR 1835643
  • 2. Yu. L. Daletskiĭ and S. V. Fomin, \cyr Mery i differentsial′nye uravneniya v beskonechnomernykh prostranstvakh, “Nauka”, Moscow, 1983 (Russian). MR 720545
  • 3. Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136
  • 4. Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374
  • 5. A. Kohatsu-Higa and P. Protter, The Euler scheme for SDE’s driven by semimartingales, Stochastic analysis on infinite-dimensional spaces (Baton Rouge, LA, 1994) Pitman Res. Notes Math. Ser., vol. 310, Longman Sci. Tech., Harlow, 1994, pp. 141–151. MR 1415665
  • 6. Alexander Kolodii, On convergence of approximations of Ito-Volterra equations, Stochastic differential and difference equations (Győr, 1996) Progr. Systems Control Theory, vol. 23, Birkhäuser Boston, Boston, MA, 1997, pp. 157–165. MR 1636834
  • 7. D. F. Kuznetsov, \cyr Nekotorye voprosy teorii chislennogo resheniya stokhasticheskikh differentsial′nykh uravneniĭ Ito, Izdatel′stvo Sankt-Peterburgskogo Gosudartsvennogo Tekhnicheskogo Universiteta, St. Petersburg, 1998 (Russian, with Russian summary). MR 1711917
  • 8. Hannelore Lisei, Approximation by time discretization of special stochastic evolution equations, Math. Pannon. 12 (2001), no. 2, 245–268. MR 1860165
  • 9. G. N. Mil′shteĭn, \cyr Chislennoe integrirovanie stokhasticheskikh differentsial′nykh uravneniĭ, Ural. Gos. Univ., Sverdlovsk, 1988 (Russian). MR 955705
    G. N. Milstein, Numerical integration of stochastic differential equations, Mathematics and its Applications, vol. 313, Kluwer Academic Publishers Group, Dordrecht, 1995. Translated and revised from the 1988 Russian original. MR 1335454
  • 10. Philip Protter and Denis Talay, The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab. 25 (1997), no. 1, 393–423. MR 1428514, https://doi.org/10.1214/aop/1024404293
  • 11. D. Talay, Simulation and numerical analysis of stochastic differential systems: a review, Probabilistic Methods in Applied Physics (P. Krée and W. Wedig, eds.), Springer, Berlin, 1995, pp. 54-96.
  • 12. Constantin Tudor and Maria Tudor, Approximation schemes for Itô-Volterra stochastic equations, Bol. Soc. Mat. Mexicana (3) 1 (1995), no. 1, 73–85. MR 1350639

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60H35, 60H10, 60H20, 65C30

Retrieve articles in all journals with MSC (2000): 60H35, 60H10, 60H20, 65C30


Additional Information

G. Shevchenko
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty of Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: zhora@univ.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-05-00625-3
Keywords: Stochastic differential equations in a Hilbert space, discrete-time approximations, equations of the It\^o--Volterra type
Received by editor(s): December 16, 2002
Published electronically: February 9, 2005
Article copyright: © Copyright 2005 American Mathematical Society