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Theory of Probability and Mathematical Statistics

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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
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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
  • 2. 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
  • 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
  • 5. Wilfried Grecksch and Constantin Tudor, Stochastic evolution equations, Mathematical Research, vol. 85, Akademie-Verlag, Berlin, 1995. A Hilbert space approach. MR 1353910
  • 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
  • 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

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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

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