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Lyapunov Functionals and Stability of Stochastic Difference Equations pp 79–108Cite as

Some Peculiarities of the Method

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Abstract

In this chapter some peculiarities of the proposed method of the construction of Lyapunov functionals are considered. It is shown in detail that this method allows us to get not sufficient stability conditions only, but the necessary and sufficient stability conditions for the trivial solution of stochastic linear difference equation too. It is shown that different ways of the estimation of the Lyapunov functional increment allow us to get different stability conditions and therefore different regions of stability. In detail, some examples of stochastic Volterra difference equations and stochastic difference equations with Markovian switching are investigated. The results of the investigations are illustrated by a lot of figures with the stability regions obtained.

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References

  1. Gikhman II, Skorokhod AV (1974) The theory of stochastic processes, vol I. Springer, Berlin

    MATH  Google Scholar 

  2. Gikhman II, Skorokhod AV (1975) The theory of stochastic processes, vol II. Springer, Berlin

    MATH  Google Scholar 

  3. Gikhman II, Skorokhod AV (1979) The theory of stochastic processes, vol III. Springer, Berlin

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  4. Mao X, Shaikhet L (2000) Delay-dependent stability criteria for stochastic differential delay equations with Markovian switching. Stab Control: Theory Appl 3(2):88–102

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  5. Shaikhet L (1996) Stability of stochastic hereditary systems with Markov switching. Theory Stoch Process 2(18)(3–4):180–185

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  6. Shaikhet L (2002) Numerical simulation and stability of stochastic systems with Markovian switching. Neural Parallel Sci. Comput. 10(2):199–208

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Authors and Affiliations

  1. Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev str. 163-A, 83015, Donetsk, Ukraine

    Leonid Shaikhet

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  1. Leonid Shaikhet
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Correspondence to Leonid Shaikhet .

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© 2011 Springer-Verlag London Limited

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Shaikhet, L. (2011). Some Peculiarities of the Method. In: Lyapunov Functionals and Stability of Stochastic Difference Equations. Springer, London. https://doi.org/10.1007/978-0-85729-685-6_5

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  • DOI: https://doi.org/10.1007/978-0-85729-685-6_5

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-684-9

  • Online ISBN: 978-0-85729-685-6

  • eBook Packages: EngineeringEngineering (R0)

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