A generalization of Mil’shtein’s theorem for stochastic differential equations
Author:
Georgiĭ Shevchenko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 78 (2009), 191-199
MSC (2000):
Primary 60H10; Secondary 34G10, 47A50, 47D06
DOI:
https://doi.org/10.1090/S0094-9000-09-00772-8
Published electronically:
August 4, 2009
MathSciNet review:
2446859
Full-text PDF Free Access
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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
- 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
- G. N. Mil′shteĭn, Chislennoe integrirovanie stokhasticheskikh differentsial′nykh uravneniĭ, Ural. Gos. Univ., Sverdlovsk, 1988 (Russian). MR 955705
- 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
- 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
- Wilfried Grecksch and Constantin Tudor, Stochastic evolution equations, Mathematical Research, vol. 85, Akademie-Verlag, Berlin, 1995. A Hilbert space approach. MR 1353910
- 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
- 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
References
- P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. MR 1214374 (94b:60069)
- G. N. Mil’shtein, Numerical Integration of Stochastic Differential Equations, Ural’skiĭ Universitet, Sverdlovsk, 1988; English transl., Kluwer, Dordrecht, 1995. MR 955705 (90k:65018); MR 1335454 (96e:65003)
- H. Schurz, Stability, Stationarity, and Boundedness of Some Implicit Numerical Methods for Stochastic Differential Equations and Applications, Logos-Verlag, Berlin, 1997. MR 1991701
- G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136 (95g:60073)
- W. Grecksch and C. Tudor, Stochastic Evolution Equations, Mathematical Research, vol. 85, Akademie-Verlag, Berlin, 1995. MR 1353910 (96m:60130)
- G. N. Mil’shtein, A theorem on the order of convergence of mean square approximations of solutions of stochastic differential equations, Teor. Veroyatnost. i Primenen. 32 (1987), no. 4, 809–811; English transl. in Theory Probab. Appl. 32 (1988), no. 4, 738–741. MR 927268 (89d:60104)
- H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, vol. 6, Pitman, Boston, MA, 1979. MR 533824 (82g:47032)
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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
Email:
zhora@univ.kiev.ua
Keywords:
Stochastic differential equation,
semilinear stochastic evolution equation,
time discretization,
Mil’shtein theorem
Received by editor(s):
January 9, 2007
Published electronically:
August 4, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
