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On the asymptotic degeneration of systems of linear inhomogeneous stochastic differential equations
Author(s):
Oleksander
Il’chenko
Translated by:
S. Kvasko
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 76
(2007).
Journal:
Theor. Probability and Math. Statist.
No. 76
(2008),
41-48.
MSC (2000):
Primary 60H10;
Secondary 34F05
Posted:
July 10, 2008
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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:
-
- 1.
- I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko, Probability Theory and Mathematical Statistics, ``Vyshcha Shkola'', Kyiv, 1979. (Russian)
- 2.
- N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland Publishing Company, Amsterdam, 1989. MR 1011252 (90m:60069)
- 3.
- 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)
- 4.
- 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)
- 5.
- 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)
- 6.
- 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)
- 7.
- 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
DOI:
10.1090/S0094-9000-08-00730-8
PII:
S 0094-9000(08)00730-8
Received by editor(s):
11/OCT/2005
Posted:
July 10, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
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