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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Barbashin-Krasovskii theorem for stochastic differential equations
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by Oleksiy Ignatyev and V. Mandrekar PDF
Proc. Amer. Math. Soc. 138 (2010), 4123-4128 Request permission

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
  • 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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 4123-4128
  • MSC (2010): Primary 60H10, 93E15
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10466-5
  • MathSciNet review: 2679634