On the properties of the second moment of solutions of stochastic differentialfunctional 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), 177184
MSC (2000):
Primary 60F15, 60G42; Secondary 62H12, 62J05
Published electronically:
August 12, 2005
MathSciNet review:
2110874
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Sufficient conditions for the mean square stability of solutions of linear stochastic differentialfunctional ItôSkorokhod 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
differentialfunctional equations under random perturbations of the
parameters, Ukrain. Mat. Ž. 25 (1973),
409–415, 432 (Russian). MR 0331514
(48 #9847)
 2.
V. E. Slyusarchuk and V. K. Yasins'ki, Stability of solutions of stochastic functionaldifferential equations in the critical case, Izv. AN BSSR Ser. fiz.mat. nauk (1977), 109115. (Russian)
 3.
A.
V. Skorokhod, Studies in the theory of random processes,
Translated from the Russian by Scripta Technica, Inc, AddisonWesley
Publishing Co., Inc., Reading, Mass., 1965. MR 0185620
(32 #3082b)
A.
V. Skorokhod, Studies in the theory of random processes,
Translated from the Russian by Scripta Technica, Inc, AddisonWesley
Publishing Co., Inc., Reading, Mass., 1965. MR 0185620
(32 #3082b)
 4.
I.
I. Gikhman and A.
V. Skorokhod, Teoriya sluchainykh protsessov. Tom III, Izdat.
“Nauka”, Moscow, 1975 (Russian). Teoriya Veroyatnostei i
Matematicheskaya Statistika. [Monographs in Probability Theory and
Mathematical Statistics]. MR 0651014
(58 #31323a)
I.
I. Gikhman and A.
V. Skorokhod, Teoriya sluchainykh protsessov. Tom III, Izdat.
“Nauka”, Moscow, 1975 (Russian). Teoriya Veroyatnostei i
Matematicheskaya Statistika. [Monographs in Probability Theory and
Mathematical Statistics]. MR 0651014
(58 #31323a)
 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 (52 #12080)
 6.
A.
N. Kolmogorov and S.
V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza,
Second edition, revised and augmented, Izdat. “Nauka”, Moscow,
1968 (Russian). MR 0234241
(38 #2559)
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
(19,44d)
 7.
Gustav
Doetsch, Handbuch der LaplaceTransformation. Band III. Anwendungen
der LaplaceTransformation, 2. Abteilung, Birkhäuser Verlag,
Basel und Stuttgart, 1956 (German). MR 0084635
(18,894c)
 8.
L.
I. Jasinskaja, Mean square stability of the trivial solution of
linear stochastic functionaldifferential equations with variable
coefficients, Ukrain. Mat. Zh. 33 (1981), no. 4,
482–488 (Russian). MR 627723
(82k:60128)
 9.
E.
A. Andreeva, V.
B. Kolmanovskiĭ, and L.
E. Shaĭkhet, Upravlenie sistemami s posledeistviem,
“Nauka”, Moscow, 1992 (Russian, with English and Russian
summaries). MR
1185708 (93i:49001)
 10.
V. K. Yasins'ki and S. V. Antonyuk, The mean square stability of differentialfunctional equations with the whole history, Proceedings of the Second International Conference on Applied Mathematics APLIMAT (February 57, 2003, Bratislava), Slovak Technical University, Bratislava, 2003, pp. 725732.
 1.
 V. E. Slyusarchuk, E. F. Tsar'kov, and V. K. Yasins'ki, The stability of the solutions of linear differentialfunctional equations under random perturbations of the parameters, Ukrain. Mat. Zh. 25 (1973), no. 3, 412418; English transl. in Ukrainian Math. J. 25 (1974), no. 3, 338343. MR 0331514 (48:9847)
 2.
 V. E. Slyusarchuk and V. K. Yasins'ki, Stability of solutions of stochastic functionaldifferential equations in the critical case, Izv. AN BSSR Ser. fiz.mat. nauk (1977), 109115. (Russian)
 3.
 A. V. Skorokhod, Studies in the Theory of Random Processes, Kiev University, Kiev, 1961; English transl., AddisonWesley Publishing Co. Inc., Reading, Mass., 1965. MR 0185620 (32:3082a); MR 0185620 (32:3082b)
 4.
 I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes, vol. 3, ``Nauka'', Moscow, 1973; English transl., SpringerVerlag, New York, 1979. MR 0651014 (58:31323a); MR 0651014 (58:31323b)
 5.
 I. I. Gikhman and I. I. Kadyrova, Certain results of the study of stochastic differential equations, Theory Random Process. 1 (1973), 5168. (Russian) MR 0391259 (52:12080)
 6.
 A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, ``Nauka'', Moscow, 1968; English transl., vol. 1, Graylock Press, Rochester, NY, 1957; vol. 2, 1961. MR 0234241 (38:2559); MR 0085462 (19:44d)
 7.
 G. Dötsch, Handbuch der LaplaceTransformation, Birkhäuser, Basel, 1956. MR 0084635 (18:894c)
 8.
 L. I. Yasinskaya, Mean square stability of the trivial solution of linear stochastic functionaldifferential equations with variable coefficients, Ukrain. Mat. Zh. 33 (1981), no. 4, 482489; English transl. in Ukrainian Math. J. 33 (1982), no. 4, 367372. MR 0627723 (82k:60128)
 9.
 E. A. Andreeva, V. B. Kolmanovski, and L. E. Shakhet, Control of Systems with Aftereffect, ``Nauka'', Moscow, 1992. (Russian) MR 1185708 (93i:49001)
 10.
 V. K. Yasins'ki and S. V. Antonyuk, The mean square stability of differentialfunctional equations with the whole history, Proceedings of the Second International Conference on Applied Mathematics APLIMAT (February 57, 2003, Bratislava), Slovak Technical University, Bratislava, 2003, pp. 725732.
Similar Articles
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:
http://dx.doi.org/10.1090/S0094900005006411
PII:
S 00949000(05)006411
Received by editor(s):
April 8, 2003
Published electronically:
August 12, 2005
Article copyright:
© Copyright 2005 American Mathematical Society
