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Stability and attraction of solutions of nonlinear stochastic differential equations with standard and fractional Brownian motions

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Abstract

Conditions have been found that provide the stability and attraction of solutions of nonlinear stochastic differential systems, with a linear deterministic part, with standard and fractional Brownian motions.

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References

  1. Izobov, N.A., Vvedenie v teoriyu pokazatelei Lyapunova (Introduction to the Theory of Lyapunov Exponents), Minsk, 2006.

    Google Scholar 

  2. Biagini, F., Hu, Y., Oksendal, B., and Zhang, T., Stochastic Calculus for Fractional Brownian Motion and Applications, London, 2008.

    Book  MATH  Google Scholar 

  3. Mishura, Y., Stochastic Calculus for Fractional Brownian Motion and Related Processes, Berlin–Heidelberg, 2008.

    Book  MATH  Google Scholar 

  4. Zahle, M., Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields, 1998, vol. 111, pp. 333–374.

    Article  MathSciNet  MATH  Google Scholar 

  5. Nualart, D. and Guerra, J., Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Stochastic Anal. Appl., 2008, vol. 25, no. 5, pp. 1053–1075.

    MathSciNet  MATH  Google Scholar 

  6. Nualart, D. and Rascanu, A., Differential equations driven by fractional Brownian motion, Collect. Math., 2002, vol. 53, no. 1, pp. 55–81.

    MathSciNet  MATH  Google Scholar 

  7. Levakov, A.A. and Vas’kovskii, M.M., Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients, Differ. Equations, 2014, vol. 50, no. 2, pp. 189–202.

    Article  MathSciNet  MATH  Google Scholar 

  8. Levakov, A.A. and Vas’kovskii, M.M., Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion, discontinuous coefficients, and a partly degenerate diffusion operator, Differ. Equations, 2014, vol. 50, no. 8, pp. 1053–1069.

    Article  MathSciNet  MATH  Google Scholar 

  9. Levakov, A.A. and Vas’kovskii, M.M., Existence of solutions of stochastic differential inclusions with standard and fractional Brownian motions, Differ. Equations, 2015, vol. 51, no. 8, pp. 991–997.

    Article  MathSciNet  MATH  Google Scholar 

  10. Khasminskii, R., Stochastic Stability of Differential Equations, London, 2012.

    Book  MATH  Google Scholar 

  11. Caraballo, T., Garrido-Atienza, M.J., and Taniguchi, T., The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal.: Theory Methods Appl., 2011, vol. 74, pp. 3671–3684.

    Article  MathSciNet  MATH  Google Scholar 

  12. Boufoussi, B. and Hajji, S., Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett., 2012, vol. 82, pp. 1549–1558.

    Article  MathSciNet  MATH  Google Scholar 

  13. Martynyuk, A.A., Lakshmikantam, V., and Lila, S., Ustoichivost’ dvizheniya: metod integral’nykh neravenstv (Stability of Motion: Method of Integral Inequalities), Kiev, 1989.

    Google Scholar 

  14. Levakov, A.A., Stokhasticheskie differentsial’nye uravneniya (Stochastic Differential Equations), Minsk, 2009.

    Google Scholar 

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Authors and Affiliations

  1. Belarusian State University, Minsk, 220030, Belarus

    M. M. Vas’kovskii

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  1. M. M. Vas’kovskii
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Correspondence to M. M. Vas’kovskii.

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Original Russian Text © M.M. Vas’kovskii, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 2, pp. 160–173.

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Vas’kovskii, M.M. Stability and attraction of solutions of nonlinear stochastic differential equations with standard and fractional Brownian motions. Diff Equat 53, 157–170 (2017). https://doi.org/10.1134/S0012266117020021

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  • DOI: https://doi.org/10.1134/S0012266117020021

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