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
Published electronically: February 9, 2005
MathSciNet review: 2110916
Full-text PDF

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. D. Barbu and V. Radu, Approximations to mild solutions of stochastic semilinear equations, Novi Sad J. Math. 30 (2000), no. 1, 183-190. MR 1835643 (2002c:60104)
  • 2. Yu. L. Daletskii and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Spaces, ``Nauka'', Moscow, 1983. (Russian) MR 0720545 (86g:46059)
  • 3. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. MR 1207136 (95g:60073)
  • 4. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. MR 1214374 (94b:60069)
  • 5. A. Kohatsu-Higa and P. Protter, The Euler scheme for SDE's driven by semimartingales, Pitman Res. Notes Math. Ser., vol. 310, Longman Sci. Tech., Harlow, 1994, pp. 141-151. MR 1415665 (97i:60074)
  • 6. A. Kolodii, On convergence of approximations of Itô-Volterra equations, Progr. Systems Control Theory, vol. 23, Birkhäuser Boston, Boston, MA, 1997, pp. 157-165. MR 1636834 (99g:60101)
  • 7. D. F. Kuznetsov, Some Problems in the Theory of Numerical Solution of Itô Stochastic Differential Equations, S.-Peterburg. Gos. Tekhn. Univ., St. Petersburg, 1998. (Russian) MR 1711917 (2000k:60116)
  • 8. H. Lisei, Approximation by time discretization of special stochastic evolution equations, Math. Pannon. 12 (2001), no. 2, 245-268. MR 1860165 (2002j:60113)
  • 9. G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Ural. Gos. Univ., Sverdlovsk, 1988; English transl., Mathematics and Its Applications, vol. 313, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 0955705 (90k:65018); MR 1335454 (96e:65003)
  • 10. P. Protter and D. Talay, The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab. 25 (1997), no. 1, 393-423. MR 1428514 (98c:60063)
  • 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. C. Tudor and M. Tudor, Approximation schemes for Itô-Volterra stochastic equations, Bol. Soc. Mat. Mexicana 1 (1995), no. 1, 73-85. MR 1350639 (96k:60165)

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

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

American Mathematical Society