Electronic Journal of Probability

Almost Sure Exponential Stability of Neutral Differential Difference Equations with Damped Stochastic Perturbations

Xiao Liao and Xuerong Mao

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In this paper we shall discuss the almost sure exponential stability for a neutral differential difference equation with damped stochastic perturbations of the form $d[x(t)-G(x(t-\tau))] = f(t,x(t),x(t-\tau))dt + \sigma(t) dw(t)$. Several interesting examples are also given for illustration. It should be pointed out that our results are even new in the case when $\sigma(t) \equiv 0$, i.e. for deterministic neutral differential difference equations.

Article information

Electron. J. Probab., Volume 1 (1996), paper no. 8, 16 pp.

Accepted: 15 April 1996
First available in Project Euclid: 25 January 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 34K20: Stability theory

neutral equations stochastic perturbation exponential martingale inequality Borel-Cantelli's lemma Lyapunov exponent

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Liao, Xiao; Mao, Xuerong. Almost Sure Exponential Stability of Neutral Differential Difference Equations with Damped Stochastic Perturbations. Electron. J. Probab. 1 (1996), paper no. 8, 16 pp. doi:10.1214/EJP.v1-8. https://projecteuclid.org/euclid.ejp/1453756471

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  • S. Albeverio, Ph. Blanchard and R. Hoegh-Krohn, A stochastic model for the orbits of planets and satellites: an interpretation of the Titius-Bode law, Expositions Mathematicae 1, (1984), 365–373.
  • R. Durran and A. Truman, Planetesimal diffusions, Preprint of University College of Swansea, 1990.
  • A. Friedman, Stochastic Differential Equations and Applications, Academic Press, Vol.1, (1975).
  • J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, (1977).
  • J. K. Hale and K. R. Meyer, A class of functional equations of neutral type, Mem. Amer. Math. Soc. 76, (1967), 1–65.
  • V. B. Kolmanovskii, On the stability of stochastic systems with delay, Problems Inform. Transmission 5(4), (1969), 59–67.
  • V. B. Kolmanovskii and V. R. Nosov, Stability and Periodic Modes of Control Systems with Aftereffect, Nauka, Moscow, (1981).
  • V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, (1986).
  • X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, (1994).
  • X. Mao, Exponential stability of large-scale stochastic differential equations, Systems and Control Letters 19, (1992), 71–81.
  • M. Metivier, Semimartingales, Walter de Gruyter, (1982).
  • S.-E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, (1986).