Parametric Models And Stochastic Integrals

  • M. Grigoriu
Conference paper
Part of the Solid Mechanics and its Applications book series (SMIA, volume 47)


Solutions of linear and nonlinear stochastic differential equations describing the response of dynamic systems with random input can be expressed as stochastic integrals involving the input and the system state vector. The stochastic integrals can be defined in the Itô or the Stratonovich sense [16]. These integrals become ordinary Stieltjes integrals in some cases if the input can be described by parametric models. Parametric models can also be used efficiently in Monte Carlo simulation studies [6,14].


Characteristic Function Gaussian Process Stochastic Differential Equation Bernstein Polynomial Random Vibration 
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  1. 1.
    Brockwell, P.J. and Davis, R.A.: Time Series: Theory and Methods, Springer-Verlag, New York, 1987.zbMATHGoogle Scholar
  2. 2.
    Gilhman, I.I. and Skorohod, A.V: Stochastic Differential Equations, Springer- Verlag, New York, 1972.Google Scholar
  3. 3.
    Grigoriu, M.: White Noise Processes in Random Vibration, International Symposium on Nonlinear Dynamics and Stochastic Mechanics, The Fields Institute for Research in Mathematical Sciences, Waterloo, Ontario, Canada, 1993.Google Scholar
  4. 4.
    Grigoriu, M.: On the spectral representation method in simulation, Probabilistic Engineering Mechanics 8 (1993), 75–90.CrossRefGoogle Scholar
  5. 5.
    Grigoriu, M. and Balopoulou, S.: A Simulation method for Stationary Gaussian Functions Based on the Sampling Theorem, Probabilistic Engineering Mechanics 8 (1993), 239–254.CrossRefGoogle Scholar
  6. 6.
    Grigoriu, M.: Applied Non-Gaussian Processes. Example, Theory, Simulation, Linear Random Vibration, and MATLAB Solutions, Prentice-Hall, Englewood Cliffs, N.J., 1995.Google Scholar
  7. 7.
    Janicki, A., Michna,Z.,and Weron, A.: Approximation of Stochastic Differential Equations Driven by a-Stable Levy Motion, Report, Hugo Steinhaus Center for Stochastic Methods, Wroclaw, Poland, 1995Google Scholar
  8. 8.
    Johnson, N.L. and Kotz, S.: Distributions in Statistics: Continuous Multivariate Distributions,John Wiley, New York, N.Y., 1972zbMATHGoogle Scholar
  9. 9.
    Naess, A. and Johnsen, J.M.: The Path Integral Solution Technique Applied to the Random Vibration of Hysteretic Systems, in P.D. Spanos and C.A. Brebbia (eds.), Computational Stochastic Mechanics, Elsevier Applied Science, London,1991Google Scholar
  10. 10.
    Parzen, E.: Stochastic Processes, Holden-Day, San Francisco, 1962.Google Scholar
  11. 11.
    Protter,P.:Stochastic Integration and Differential Equation, Springer-Verlag, New York,N.Y.,1990.Google Scholar
  12. 12.
    Rosinski, J. and Woyczynski,W.A.: On Itö Stochastic Integration with respect to p-Stable Motion: Inner Clock, Integrability of Sample Paths, Double and Multiple Integration, The Annals of Probability 14 (1986), 271–286MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Samorodnitsky, G. and Taqqu, M.S.: Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance, Chapman and Hall, New York, N.Y., 1994zbMATHGoogle Scholar
  14. 14.
    Shinozuka, M. and Deodatis, G.: Simulation of Stochastic Processes by Spectral Representation,Applied Mechanics Reviews 44 (1991),191–203MathSciNetCrossRefGoogle Scholar
  15. 15.
    Soize, C.:The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, World Scientific, Singapore, 1994zbMATHGoogle Scholar
  16. 16.
    Soong, T.T. and Grigoriu, M.: Random Vibration of Mechanical and Structural Systems, Prentice-Hall, Englewood Cliffs, N.J., 1993.Google Scholar
  17. 17.
    Wong, E. and Hajek, B.: Stochastic Processes in Engineering Systems, Springer- Verlag, New York,N.Y., 1985.zbMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • M. Grigoriu
    • 1
  1. 1.Cornell UniversityIthacaUSA

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