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

DOI:
https://doi.org/10.1090/S0094-9000-09-00783-2

Published electronically:
December 30, 2009

Full-text PDF

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.

**1.**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)****2.**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****3.**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)****4.**R. Bellman and K. L. Cooke,*Differential-Difference Equations*, Academic Press, New York-London, 1967. MR**0147745 (26:5259)****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.**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)****7.**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)****8.**V. B. Kolmanovskiĭ and V. R. Nosov,*Stability and Periodic Modes of Adaptable Systems with Aftereffect*, Nauka, Moscow, 1981. (Russian) MR**641554 (83g:34001)****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.**1**(1998), no. 5, 107-112, 158. (Russian) MR**1700664****11.**R. Horn and C. Johnson,*Matrix Analysis*, Cambridge University Press, Cambridge, 1985. MR**832183 (87e:15001)****12.**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)****13.**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)****14.**E. F. Tsar'kov,*Random Perturbations of Differential-Functional Equations*, Zinatne, Riga, 1989. (Russian) MR**1036733 (90m:34164)****15.**J. Hale,*Theory of Functional Differential Equations*, Springer, Berlin, 1978. MR**0508721 (58:22904)****16.**I. I. Gihman and A. V. Skorohod,*Stochastic Differential Equations*, Springer, Berlin, 1972. MR**0346904 (49:11625)**

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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

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

DOI:
https://doi.org/10.1090/S0094-9000-09-00783-2

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