On the properties of the second moment of solutions of stochastic differential-functional equations with varying coefficients

Authors:
V. K. Yasins'kii and S. V. Antonyuk

Translated by:
V. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **70** (2004).

Journal:
Theor. Probability and Math. Statist. **70** (2005), 177-184

MSC (2000):
Primary 60F15, 60G42; Secondary 62H12, 62J05

DOI:
https://doi.org/10.1090/S0094-9000-05-00641-1

Published electronically:
August 12, 2005

MathSciNet review:
2110874

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient conditions for the mean square stability of solutions of linear stochastic differential-functional Itô-Skorokhod equations with unbounded aftereffect are obtained in the paper. The critical case is also studied.

**1.**V. E. Sljusarčuk, E. F. Car′kov, and V. K. Jasinskiĭ,*The stability of the solutions of linear differential-functional equations under random perturbations of the parameters*, Ukrain. Mat. Ž.**25**(1973), 409–415, 432 (Russian). MR**0331514****2.**V. E. Slyusarchuk and V. K. Yasins'ki,*Stability of solutions of stochastic functional-differential equations in the critical case*, Izv. AN BSSR Ser. fiz.-mat. nauk (1977), 109-115. (Russian)**3.**A. V. Skorokhod,*Studies in the theory of random processes*, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0185620****4.**I. I. Gikhman and A. V. Skorokhod,*\cyr Teoriya sluchaĭnykh protsessov. Tom III.*, Izdat. “Nauka”, Moscow, 1975 (Russian). \cyr Teoriya Veroyatnosteĭ i Matematicheskaya Statistika. [Monographs in Probability Theory and Mathematical Statistics]. MR**0651014****5.**Ĭ. Ī. Gīhman and I. I. Kadyrova,*Certain results of the study of stochastic differential equations*, Theory of random processes, No. 1 (Russian), Izdat. “Naukova Dumka”, Kiev, 1973, pp. 51–68, 140 (Russian). MR**0391259****6.**A. N. Kolmogorov and S. V. Fomin,*\cyrÈlementy teorii funktsiĭ i funktsional′nogo analiza*, Second edition, revised and augmented, Izdat. “Nauka”, Moscow, 1968 (Russian). MR**0234241**

A. N. Kolmogorov and S. V. Fomin,*Elements of the theory of functions and functional analysis. Vol. 1. Metric and normed spaces*, Graylock Press, Rochester, N. Y., 1957. Translated from the first Russian edition by Leo F. Boron. MR**0085462****7.**Gustav Doetsch,*Handbuch der Laplace-Transformation. Band III. Anwendungen der Laplace-Transformation, 2. Abteilung*, Birkhäuser Verlag, Basel und Stuttgart, 1956 (German). MR**0084635****8.**L. I. Yasinskaya,*Mean square stability of the trivial solution of linear stochastic functional-differential equations with variable coefficients*, Ukrain. Mat. Zh.**33**(1981), no. 4, 482-489; English transl. in Ukrainian Math. J.**33**(1982), no. 4, 367-372. MR**0627723 (82k:60128)****9.**E. A. Andreeva, V. B. Kolmanovskiĭ, and L. E. Shaĭkhet,*\cyr Upravlenie sistemami s posledeĭstviem*, “Nauka”, Moscow, 1992 (Russian, with English and Russian summaries). MR**1185708****10.**V. K. Yasins'ki and S. V. Antonyuk,*The mean square stability of differential-functional equations with the whole history*, Proceedings of the Second International Conference on Applied Mathematics APLIMAT (February 5-7, 2003, Bratislava), Slovak Technical University, Bratislava, 2003, pp. 725-732.

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2000):
60F15,
60G42,
62H12,
62J05

Retrieve articles in all journals with MSC (2000): 60F15, 60G42, 62H12, 62J05

Additional Information

**V. K. Yasins'kii**

Affiliation:
Department of Mathematics, Chernivtsi National Yuriy Fedkovich University, Universitets’ka Street 28, Chernivtsi 58012, Ukraine

Email:
yasik@cv.ukrtel.net

**S. V. Antonyuk**

Affiliation:
Department of Mathematics, Chernivtsi National Yuriy Fedkovich University, Universitets’ka Street 28, Chernivtsi 58012, Ukraine

DOI:
https://doi.org/10.1090/S0094-9000-05-00641-1

Received by editor(s):
April 8, 2003

Published electronically:
August 12, 2005

Article copyright:
© Copyright 2005
American Mathematical Society