Wide sense stationary solutions of difference equations in a Banach space
Author:
M. F. Gorodniĭ
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 74 (2007), 29-35
MSC (2000):
Primary 60G10, 39A10; Secondary 47A50
DOI:
https://doi.org/10.1090/S0094-9000-07-00695-3
Published electronically:
June 25, 2007
MathSciNet review:
2336776
Full-text PDF Free Access
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Additional Information
Abstract: A criterion is proved for the existence of a unique wide sense stationary solution of a linear difference equation with operator coefficients in a Banach space. The stability of this solution with respect to small perturbations of operator coefficients is proved.
References
- M. F. Gorodnīĭ, Solutions of a stochastic difference equation that are stationary and bounded in the mean, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 8 (2002), 12–16 (Ukrainian, with English summary). MR 2009511
- S. Bochner and R. S. Phillips, Absolutely convergent Fourier expansions for non-commutative normed rings, Ann. of Math. (2) 43 (1942), 409–418. MR 7939, DOI https://doi.org/10.2307/1968800
- M. F. Gorodniĭ, Bounded and periodic solutions of a difference equation and its stochastic analogue in a Banach space, Ukrain. Mat. Zh. 43 (1991), no. 1, 41–46 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J. 43 (1991), no. 1, 32–37. MR 1098269, DOI https://doi.org/10.1007/BF01066900
- V. M. Kruglov, Dopolnitel′nye glavy teorii veroyatnosteĭ, “Vyssh. Shkola”, Moscow, 1984 (Russian). MR 756812
- L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
- A. Ya. Dorogovtsev, Stability of periodic solutions of operator equations with perturbation coefficients, Exploring stochastic laws, VSP, Utrecht, 1995, pp. 111–119. MR 1713997
- A. Ya. Dorogovtsev, Stability of bounded and stationary solutions of linear equations with respect to perturbations of operator coefficients, Dokl. Akad. Nauk 345 (1995), no. 4, 448–450 (Russian). MR 1376444
References
- M. F. Gorodniĭ, Solutions of a stochastic difference equation that are stationary and bounded in the mean, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 8 (2002), 12–16. MR 2009511 (2004g:60099)
- S. Bochner and R. S. Phillips, Absolutely convergent Fourier expansions for non-commutative normed rings, Ann. Math. 43 (1942), no. 3, 409–418. MR 0007939 (4:218g)
- M. F. Gorodniĭ, Bounded and periodic solutions of a difference equation and its stochastic analogue in a Banach space, Ukrain. Mat. Zh. 43 (1991), no. 1, 41–46; English transl. in Ukrainian Math. J. 43 (1991), no. 1, 32–37. MR 1098269 (92d:60068)
- V. M. Kruglov, Supplementary Chapters on Probability Theory, “Vysshaya Shkola”, Moscow, 1984. (Russian) MR 756812 (86d:60009)
- L. V. Kantorovich and G. P. Akilov, Functional Analysis, “Nauka”, Moscow, 1984; English transl., Pergamon Press, Oxford-Elmsford, N.Y., 1982. MR 664597 (83h:46002)
- A. Ya. Dorogovtsev, Stability of periodic solutions of operator equations with perturbation coefficients, Exploring Stochastic Laws (A. V. Skorokhod and Yu. V. Borovskikh, eds.), VSP, Utrecht, The Netherlands, 1995, pp. 111–119. MR 1713997 (2001g:60163)
- A. Ya. Dorogovtsev, Stability of bounded and stationary solutions of linear equations with respect to perturbations of operator coefficients, Dokl. Akad. Nauk 345 (1995), no. 4, 448–450. MR 1376444 (96k:47123)
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Additional Information
M. F. Gorodniĭ
Affiliation:
Department of Mathematical Analysis, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Glushkov Avenue, 6, Kyiv, 03127, Ukraine
Email:
gorodnii@yandex.ru
Keywords:
Difference equation,
operator coefficients,
wide sense stationary solutions
Received by editor(s):
March 18, 2005
Published electronically:
June 25, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
