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

DOI:
https://doi.org/10.1090/S0002-9939-2010-10466-5

Published electronically:
July 7, 2010

MathSciNet review:
2679634

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Abstract | References | Similar Articles | Additional Information

Abstract: A system of stochastic differential equations which has a zero solution is considered. It is assumed that there exists a positive definite function such that the corresponding operator is nonpositive. It is proved that if the set 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:
https://doi.org/10.1090/S0002-9939-2010-10466-5

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.