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A generalization of Mil'shtein's theorem for stochastic differential equations
Author(s):
Georgiĭ
Shevchenko
Translated by:
S. Kvasko
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 78
(2008).
Journal:
Theor. Probability and Math. Statist.
No. 78
(2009),
191-199.
MSC (2000):
Primary 60H10;
Secondary 34G10, 47A50, 47D06
Posted:
August 4, 2009
MathSciNet review:
2446859
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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:
-
- 1.
- P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. MR 1214374 (94b:60069)
- 2.
- 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)
- 3.
- H. Schurz, Stability, Stationarity, and Boundedness of Some Implicit Numerical Methods for Stochastic Differential Equations and Applications, Logos-Verlag, Berlin, 1997. MR 1991701
- 4.
- 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)
- 5.
- W. Grecksch and C. Tudor, Stochastic Evolution Equations, Mathematical Research, vol. 85, Akademie-Verlag, Berlin, 1995. MR 1353910 (96m:60130)
- 6.
- 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)
- 7.
- 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
DOI:
10.1090/S0094-9000-09-00772-8
PII:
S 0094-9000(09)00772-8
Keywords:
Stochastic differential equation,
semilinear stochastic evolution equation,
time {\discretisation },
Mil'shtein theorem
Received by editor(s):
9/JAN/2007
Posted:
August 4, 2009
Copyright of article:
Copyright
2009,
American Mathematical Society
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