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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Barbashin-Krasovskii theorem for stochastic differential equations


Authors: Oleksiy Ignatyev and V. Mandrekar
Journal: Proc. Amer. Math. Soc. 138 (2010), 4123-4128
MSC (2010): Primary 60H10, 93E15
Published electronically: July 7, 2010
MathSciNet review: 2679634
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Abstract: A system of stochastic differential equations $ dX(t)=f(X)dt+ \sum_{i=1}^{k}g_i(X)dW_i(t)$ which has a zero solution $ X=0$ is considered. It is assumed that there exists a positive definite function $ V(x)$ such that the corresponding operator $ LV$ is nonpositive. It is proved that if the set $ \{M:~ LV=0\}$ does not include entire semitrajectories of the system almost surely, then the zero solution is asymptotically stable in probability.


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Additional Information

Oleksiy Ignatyev
Affiliation: Department of Statistics and Probability, Michigan State University, A408 Wells Hall, East Lansing, Michigan 48824-1027
Email: ignatyev@stt.msu.edu

V. Mandrekar
Affiliation: Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824-1027
Email: mandrekar@stt.msu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10466-5
PII: S 0002-9939(2010)10466-5
Keywords: Stochastic differential equations, Lyapunov functions, asymptotic stability in probability
Received by editor(s): August 7, 2009
Received by editor(s) in revised form: February 27, 2010
Published electronically: July 7, 2010
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.