On the properties of the second moment of solutions of stochastic differential-functional equations with varying coefficients
Authors:
V. K. Yasins’kiĭ and S. V. Antonyuk
Translated by:
V. Semenov
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 Free Access
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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.
References
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- V. E. Slyusarchuk and V. K. Yasins’kiĭ, Stability of solutions of stochastic functional-differential equations in the critical case, Izv. AN BSSR Ser. fiz.-mat. nauk (1977), 109–115. (Russian)
- A. V. Skorokhod, Studies in the theory of random processes, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. Translated from the Russian by Scripta Technica, Inc. MR 0185620
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- Gustav Doetsch, Handbuch der Laplace-Transformation. Band III. Anwendungen der Laplace-Transformation, 2. Abteilung, Birkhäuser Verlag, Basel und Stuttgart, 1956 (German). MR 0084635
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- E. A. Andreeva, V. B. Kolmanovskiĭ, and L. E. Shaĭkhet, Upravlenie sistemami s posledeĭ stviem, “Nauka”, Moscow, 1992 (Russian, with English and Russian summaries). MR 1185708
- V. K. Yasins’kiĭ 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.
References
- V. E. Slyusarchuk, E. F. Tsar’kov, and V. K. Yasins’kiĭ, The stability of the solutions of linear differential-functional equations under random perturbations of the parameters, Ukrain. Mat. Zh. 25 (1973), no. 3, 412–418; English transl. in Ukrainian Math. J. 25 (1974), no. 3, 338–343. MR 0331514 (48:9847)
- V. E. Slyusarchuk and V. K. Yasins’kiĭ, Stability of solutions of stochastic functional-differential equations in the critical case, Izv. AN BSSR Ser. fiz.-mat. nauk (1977), 109–115. (Russian)
- A. V. Skorokhod, Studies in the Theory of Random Processes, Kiev University, Kiev, 1961; English transl., Addison-Wesley Publishing Co. Inc., Reading, Mass., 1965. MR 0185620 (32:3082a); MR 0185620 (32:3082b)
- I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes, vol. 3, “Nauka”, Moscow, 1973; English transl., Springer-Verlag, New York, 1979. MR 0651014 (58:31323a); MR 0651014 (58:31323b)
- I. I. Gikhman and I. I. Kadyrova, Certain results of the study of stochastic differential equations, Theory Random Process. 1 (1973), 51–68. (Russian) MR 0391259 (52:12080)
- 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)
- G. Dötsch, Handbuch der Laplace-Transformation, Birkhäuser, Basel, 1956. MR 0084635 (18:894c)
- 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)
- E. A. Andreeva, V. B. Kolmanovskiĭ, and L. E. Shaĭkhet, Control of Systems with Aftereffect, “Nauka”, Moscow, 1992. (Russian) MR 1185708 (93i:49001)
- V. K. Yasins’kiĭ 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.
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Additional Information
V. K. Yasins’kiĭ
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
Received by editor(s):
April 8, 2003
Published electronically:
August 12, 2005
Article copyright:
© Copyright 2005
American Mathematical Society