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
Journal:
Theor. Probability and Math. Statist. 79 (2009), 89-100
MSC (2000):
Primary 60F15; Secondary 60G44
DOI:
https://doi.org/10.1090/S0094-9000-09-00783-2
Published electronically:
December 30, 2009
MathSciNet review:
2494538
Full-text PDF Free Access
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.
References
- N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Vvedenie v teoriyu funktsional′no-differentsial′nykh uravneniĭ, “Nauka”, Moscow, 1991 (Russian). With an English summary. MR 1144998
- 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
- 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
- Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 0147745
- 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)
- 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
- 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
- V. B. Kolmanovskiĭ and V. R. Nosov, Ustoĭ chivost′i periodicheskie rezhimy reguliruemykh sistem s posledeĭ stviem, “Nauka”, Moscow, 1981 (Russian). Teoreticheskie Osnovy Tekhnicheskoĭ Kibernetiki. [Theoretical Foundations of Engineering Cybernetics]. MR 641554
- V. Yu. Slyusarchuk, The Absolute Stability of Dynamical Systems with Aftereffect, UDUVGP, Rivne, 2003. (Ukrainian)
- 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
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183
- 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, DOI https://doi.org/10.1007/BF01089203
- D. Ya. Khusainov and A. V. Shatyrko, Metod funktsiĭ Lyapunova v issledovanii ustoĭ chivosti differentsial′no-funktsional′nykh sistem, Izdatel′stvo Kievskogo Universiteta, Kiev, 1997 (Russian, with Russian summary). MR 1486825
- E. F. Tsar′kov, Sluchaĭ nye vozmushcheniya differentsial′no-funktsional′nykh uravneniĭ, “Zinatne”, Riga, 1989 (Russian). MR 1036733
- Jack Hale, Theory of functional differential equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 3. MR 0508721
- 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
References
- N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Linear Functional-Differential Equations, Nauka, Moscow, 1991; English transl., World Federation Publishers Company, Atlanta, GA, 1995. MR 1144998 (92j:34123); MR 1375462 (96m:34132)
- E. A. Andreeva, V. B. Kolmanovskiĭ, and L. E. Shaĭkhet, Control of Systems with Aftereffect, Nauka, Moscow, 1992; English transl., V. B. Kolmanovskiĭ and L. E. Shaĭkhet, American Mathematical Society, Providence, RI, 1996. MR 1185708 (93i:49001); MR 1415834
- R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ, The theory of equations of neutral type, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Mathematical Analysis 19 (1982), no. 1, 55–126, 232. (Russian) MR 657948 (83j:34078)
- R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York–London, 1967. MR 0147745 (26:5259)
- 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)
- G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, New York–Heidelberg, 1974; Translated from the second German edition. MR 0344810 (49:9549)
- J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Nauka, Moscow, 1994; English transl. of the second edition, Springer-Verlag, Berlin, 2003. MR 1943877 (2003j:60001)
- V. B. Kolmanovskiĭ and V. R. Nosov, Stability and Periodic Modes of Adaptable Systems with Aftereffect, Nauka, Moscow, 1981. (Russian) MR 641554 (83g:34001)
- V. Yu. Slyusarchuk, The Absolute Stability of Dynamical Systems with Aftereffect, UDUVGP, Rivne, 2003. (Ukrainian)
- I. Ya. Spektorskiĭ, A generalization of the constant variation formula for a linear nonhomogeneous stochastic equation, Problemy Upravlen. Inform. 1 (1998), no. 5, 107–112, 158. (Russian) MR 1700664
- R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. MR 832183 (87e:15001)
- D. Ya. Khusainov, Estimates for the stability of solutions of systems of functional-differential equations of neutral type, Ukrain. Mat. Zh. 1 (1991), no. 9, 1123–1135; English transl. in Ukrainian Math. J. 1 (1991), no. 9, 1053–1063. MR 1149573 (92k:34102)
- D. Ya. Khusainov and A. V. Shatyrko, The method of Lyapunov functions in the studies of the stability of differential-functional systems, Kiev University, Kiev, 1997. (Russian) MR 1486825 (2000k:34124)
- E. F. Tsar’kov, Random Perturbations of Differential-Functional Equations, Zinatne, Riga, 1989. (Russian) MR 1036733 (90m:34164)
- J. Hale, Theory of Functional Differential Equations, Springer, Berlin, 1978. MR 0508721 (58:22904)
- I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer, Berlin, 1972. MR 0346904 (49:11625)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
60F15,
60G44
Retrieve articles in all journals
with MSC (2000):
60F15,
60G44
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
Email:
M_Ihor_V@rambler.ru
E. F. Tsar’kov
Affiliation:
Department of Probability Theory and Mathematical Statistics, Riga Technical University, Riga, Latvia
Email:
carkovs@livas.lv
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
Email:
yasinsk@cv.ukrtel.net
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
