On the asymptotic degeneration of systems of linear inhomogeneous stochastic differential equations
Author:
Oleksander Il’chenko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 76 (2008), 41-48
MSC (2000):
Primary 60H10; Secondary 34F05
DOI:
https://doi.org/10.1090/S0094-9000-08-00730-8
Published electronically:
July 10, 2008
MathSciNet review:
2368738
Full-text PDF Free Access
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Abstract: Assuming the almost sure stability of a linear homogeneous system, we obtain sufficient conditions for the convergence to zero, in probability as well as pathwise, of solutions of the system of linear inhomogeneous stochastic differential equations.
References
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- Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
- O. Īl′chenko, Stochastically bounded solutions of a linear inhomogeneous stochastic differential equation, Teor. Ĭmovīr. Mat. Stat. 68 (2003), 37–43 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 68 (2004), 41–48. MR 2000393, DOI https://doi.org/10.1090/S0094-9000-04-00596-4
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- A. M. Sadovjak and E. F. Car′kov, An analogue of the Cauchy formula for stochastic differential equations, Teor. Verojatnost. i Primenen. 18 (1973), 415–417 (Russian, with English summary). MR 0319270
- A. V. Skorokhod, Asymptotic methods in the theory of stochastic differential equations, Translations of Mathematical Monographs, vol. 78, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 1020057
References
- I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko, Probability Theory and Mathematical Statistics, “Vyshcha Shkola”, Kyiv, 1979. (Russian)
- N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland Publishing Company, Amsterdam, 1989. MR 1011252 (90m:60069)
- A. V. Il’chenko, Stochastically bounded solutions of a linear inhomogeneous stochastic differential equation, Teor. Imovir. Mat. Stat. 68 (2003), 37–43; English transl. in Theory Probab. Math. Statist. 68 (2004), 48–55. MR 2000393 (2004d:60150)
- A. V. Il’chenko, Stochastically bounded solutions of the linear inhomogeneous stochastic differential equation system, Theory Stoch. Process. 9(25) (2003), no. 1–2, 65–72. MR 2080014 (2005h:60166)
- R. Z. Khasminskiĭ, Stochastic Stability of Differential Equations, “Nauka”, Moscow, 1969; English transl., Sijthoff & Noordhoff, Alphen aan den Rijn–Germantown, Md., 1980. MR 600653 (82b:60064)
- A. M. Sadovyak and E. F. Car’kov, An analogue of the Cauchy formula for stochastic differential equations, Teor. Veroyatnost. i Primenen. 18 (1973), no. 2, 415–417; English transl. in Theory Probab. Appl. 18 (1973), no. 2, 394–396. MR 0319270 (47:7814)
- A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, “Naukova dumka”, Kiev, 1987; English transl., American Mathematical Society, Providence, RI, 1989. MR 1020057 (90i:60038)
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Additional Information
Oleksander Il’chenko
Affiliation:
Department of General Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Received by editor(s):
October 11, 2005
Published electronically:
July 10, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
