\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Khasminskii-type theorems for stochastic functional differential equations

Abstract Related Papers Cited by
  • For a stochastic functional differential equation (SFDE) to have a unique global solution it is in general required that the coefficients of the SFDE obey the local Lipschitz condition and the linear growth condition. However, there are many SFDEs in practice which do not obey the linear growth condition. The main aim of this paper is to establish existence-and-uniqueness theorems for SFDEs where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Laypunov-type functions.
    Mathematics Subject Classification: 34K50, 60H10, 34F05, 93E03.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl., 292 (2004), 364-380.doi: 10.1016/j.jmaa.2003.12.004.

    [2]

    A. Bahar and X. Mao, Stochastic delay population dynamics, International J. Pure and Applied Math., 11 (2004), 377-400.

    [3]

    T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988.

    [4]

    R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980.

    [5]

    V. Kolmanovskiĭ and A. Myshkis, "Applied Theory of Functional-Differential Equations," Mathematics and its Applications (Soviet Series), 85, Kluwer Academic Publishers Group, Dordrecht, 1992.

    [6]

    G. S. Ladde and V. Lakshmikantham, "Random Differential Inequalities," Mathematics in Science and Engineering, 150, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.

    [7]

    M. Loève, "Probability Theory," Third edition, D. Van Nostrand Company, Inc., Princeton, N. J.-Toronto, Ont.-London, 1963.

    [8]

    Q. Luo, X. Mao and Y. Shen, Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations, Automatica J. IFAC, 47 (2011), 2075-2081.doi: 10.1016/j.automatica.2011.06.014.

    [9]

    X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales," Pitman Research Notes in Mathematics Series, 251, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991.

    [10]

    X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994.

    [11]

    X. Mao, "Stochastic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chichester, 2008.

    [12]

    X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002), 125-142.doi: 10.1006/jmaa.2001.7803.

    [13]

    X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2005), 1045-1069.doi: 10.1080/07362990500118637.

    [14]

    X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching," Imperial College Press, London, 2006.

    [15]

    X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations in population dynamics, J. Math. Anal. Appl., 304 (2005), 296-320.doi: 10.1016/j.jmaa.2004.09.027.

    [16]

    S. E. A. Mohammed, "Stochastic Functional Differential Equations," Research Notes in Mathematics, 99, Pitman (Advanced Publishing Program), Boston, MA, 1984.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(397) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return