Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
Mobile Device Pairing
Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)


A generalization of Mil'shtein's theorem for stochastic differential equations

Author: Georgiĭ Shevchenko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 191-199
MSC (2000): Primary 60H10; Secondary 34G10, 47A50, 47D06
Published electronically: August 4, 2009
MathSciNet review: 2446859
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A theorem on a relationship between local and global rates of convergence for a stochastic differential equation is proved in the paper. This theorem implies the convergence of Euler type approximations for semilinear evolution equations with a coercive operator in a Hilbert space.

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

  • 1. 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 (94b:60069)
  • 2. G. N. Mil′shteĭn, Chislennoe integrirovanie stokhasticheskikh differentsialnykh uravnenii, Ural. Gos. Univ., Sverdlovsk, 1988 (Russian). MR 955705 (90k:65018)
    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 (96e:65003)
  • 3. Henri Schurz, Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications, Logos Verlag Berlin, Berlin, 1997 (English, with English and German summaries). MR 1991701
  • 4. 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 (95g:60073)
  • 5. Wilfried Grecksch and Constantin Tudor, Stochastic evolution equations, Mathematical Research, vol. 85, Akademie-Verlag, Berlin, 1995. A Hilbert space approach. MR 1353910 (96m:60130)
  • 6. G. N. Mil′shteĭn, A theorem on the order of convergence of mean-square approximations of solutions of systems of stochastic differential equations, Teor. Veroyatnost. i Primenen. 32 (1987), no. 4, 809–811 (Russian). MR 927268 (89d:60104)
  • 7. Hiroki Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. Translated from the Japanese by N. Mugibayashi and H. Haneda. MR 533824 (82g:47032)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60H10, 34G10, 47A50, 47D06

Retrieve articles in all journals with MSC (2000): 60H10, 34G10, 47A50, 47D06

Additional Information

Georgiĭ Shevchenko
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty of Mechanics and Mathematics, Kiev National Taras Shevchenko University, 64 Volodymyrska Street, 01033 Kiev, Ukraine

PII: S 0094-9000(09)00772-8
Keywords: Stochastic differential equation, semilinear stochastic evolution equation, time {\discretisation }, Mil'shtein theorem
Received by editor(s): January 9, 2007
Published electronically: August 4, 2009
Article copyright: © Copyright 2009 American Mathematical Society