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Asymptotic behavior of the solution of a linear stochastic differential-difference equation of neutral type

Authors: I. V. Malyk, E. F. Tsar'kov and V. K. Yasyns'kyĭ
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 79 (2008).
Journal: Theor. Probability and Math. Statist. 79 (2009), 89-100
MSC (2000): Primary 60F15; Secondary 60G44
Published electronically: December 30, 2009
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Abstract | References | Similar Articles | Additional Information

Abstract: Necessary and sufficient conditions are found for the exponential mean square stability of a stochastic differential-difference linear equation of neutral type in the scalar case.

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  • 1. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, \cyr Vvedenie v teoriyu funktsional′no-differentsial′nykh uravneniĭ, “Nauka”, Moscow, 1991 (Russian). With an English summary. MR 1144998
    N. V. Azbelev and L. F. Rakhmatullina, On extension of the Vallée-Poussin theorem to equations with aftereffect, Boundary value problems for functional-differential equations, World Sci. Publ., River Edge, NJ, 1995, pp. 23–36. MR 1375462
  • 2. 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
    V. B. Kolmanovskiĭ and L. E. Shaĭkhet, Control of systems with aftereffect, Translations of Mathematical Monographs, vol. 157, American Mathematical Society, Providence, RI, 1996. Translated from Control of systems with aftereffect (Russian) [“Nauka”, Moscow, 1992; MR1185708 (93i:49001)] by Victor Kotov. MR 1415834
  • 3. R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ, The theory of equations of neutral type, Mathematical analysis, Vol. 19, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1982, pp. 55–126, 232 (Russian). MR 657948
  • 4. Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 0147745
  • 5. Yu. V. Bereza and V. K. Yasyns'kyĭ, The existence of solutions of stochastic differential functional equations of the neutral type with Poissonian switchings, Visn. Kyiv. Univ., Ser. Fiz.-Mat. Nauky 5 (2002), no. 1, 19-27. (Ukrainian)
  • 6. Gustav Doetsch, Introduction to the theory and application of the Laplace transformation, Springer-Verlag, New York-Heidelberg, 1974. Translated from the second German edition by Walter Nader. MR 0344810
  • 7. Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 2003. MR 1943877
  • 8. V. B. Kolmanovskiĭ and V. R. Nosov, \cyr Ustoĭchivost′ i periodicheskie rezhimy reguliruemykh sistem s posledeĭstviem, “Nauka”, Moscow, 1981 (Russian). \cyr Teoreticheskie Osnovy Tekhnicheskoĭ Kibernetiki. [Theoretical Foundations of Engineering Cybernetics]. MR 641554
  • 9. V. Yu. Slyusarchuk, The Absolute Stability of Dynamical Systems with Aftereffect, UDUVGP, Rivne, 2003. (Ukrainian)
  • 10. I. Ya. Spektorskiĭ, A generalization of the constant variation formula for a linear nonhomogeneous stochastic equation, Problemy Upravlen. Inform. 5 (1998), 107–112, 158 (Russian, with English, Russian and Ukrainian summaries). MR 1700664
  • 11. Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183
  • 12. D. Ya. Khusainov, Estimates for the stability of solutions of systems of functional-differential equations of neutral type, Ukrain. Mat. Zh. 43 (1991), no. 9, 1123–1135 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J. 43 (1991), no. 9, 1053–1063 (1992). MR 1149573,
  • 13. D. Ya. Khusainov and A. V. Shatyrko, \cyr Metod funktsiĭ Lyapunova v issledovanii ustoĭchivosti differentsial′no-funktsional′nykh sistem, Izdatel′stvo Kievskogo Universiteta, Kiev, 1997 (Russian, with Russian summary). MR 1486825
  • 14. E. F. Tsar′kov, \cyr Sluchaĭnye vozmushcheniya differentsial′no-funktsional′nykh uravneniĭ, “Zinatne”, Riga, 1989 (Russian). MR 1036733
  • 15. Jack Hale, Theory of functional differential equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 3. MR 0508721
  • 16. Ĭ. Ī. Gīhman and A. V. Skorohod, Stochastic differential equations, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by Kenneth Wickwire; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. MR 0346904

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Additional Information

I. V. Malyk
Affiliation: Department of Mathematical and Applied Statistics, Faculty for Applied Mathematics, Chernivtsi Fed’kovych National University, Kotsyubyns’kyi Street 2, Chernivtsi 58000, Ukraine

E. F. Tsar'kov
Affiliation: Department of Probability Theory and Mathematical Statistics, Riga Technical University, Riga, Latvia

V. K. Yasyns'kyĭ
Affiliation: Department of Mathematical and Applied Statistics, Faculty for Applied Mathematics, Chernivtsi Fed’kovych National University, Kotsyubyns’kyi Street 2, Chernivtsi 58000, Ukraine

Keywords: Stochastic differential equations of neutral type, difference equations, eigenvalues
Received by editor(s): February 20, 2008
Published electronically: December 30, 2009
Article copyright: © Copyright 2009 American Mathematical Society