Approximation of solutions to differential equations with random inputs by diffusion processes

  • Harold J. Kushner
Part I: Survey Lectures
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 16)


Let yɛ(·) denote a random process whose bandwidth, loosely speaking, goes to ∞ as ɛ → 0. Consider the family of differential equations xɛ=g(xɛ,yɛ)+f(xɛ,yɛ)/α(ɛ), where α(ɛ) → 0 as ɛ → 0. The question of interest is: does the sequence {xɛ(·)} converge in some sense and if so which, if any, ordinary or Itô differential equation does it satisfy? Normally, the limit is taken in the sense of weak convergence. The problem is of great practical importance, for such questions arise in many practical situations arising in many fields. Often the limiting equation is nice and can be treated much more easily than can the xɛ(·). In any case, in practice approximations to properties of the xɛ(·) are usually sought in terms of ɛ and some limit. To illustrate these points, as well as a related stability problem, we give a practical example which arises in the theory of adaptive arrays of antennas.

The topic of convergence has seen much work, starting with the fundamental papers of Wong and Zakai, and followed by others, including Khazminskii, Papanicolaou and Kohler, etc. From a non-probabilistic point of view, it has been dealt with by McShane and Sussmann. In this paper, we discuss a rather general and efficient method of getting the correct limits. The idea exploits some general semigroup approximation results of Kurtz, and often not only gets better results than those obtained by preceding methods, but is also easier to use.


Stochastic Differential Equation Weak Convergence Automatic Gain Control Adaptive Array Finite Dimensional Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    E. Wong and M. Zakai, "On the relationship between ordinary and stochastic differential equations", Int. J. Engin. Science, (1965), 3, 213–229.Google Scholar
  2. [2]
    E. Wong and M. Zakai, "On the convergence of ordinary integrals to stochastic integrals", Ann. Math. Statist. 36, 1560–1564.Google Scholar
  3. [3]
    R. Z. Khasminskii, "A limit theorem for solutions of differential equations with random right hand sides", Theory of Prob. and Applic., (1966), 11, 390–406.Google Scholar
  4. [4]
    G. C. Papanicolaou and W. Kohler, "Asymptotic theory of mixing stochastic ordinary differential equations", Comm. Pure and Appl. Math., (1974), 27, 641–668.Google Scholar
  5. [5]
    G. C. Papanicolaou, "Some probabilistic problems and methods in singular perturbations", Rocky Mountain Journal of Math., (1976), 6, 653–674.Google Scholar
  6. [6]
    G. Blankenship and G. C. Papanicolaou, "Stability and control of stochastic systems with wide-band noise disturbances", SIAM J. on Appl. Math., (1978), 34, 437–476.Google Scholar
  7. [7]
    H. J. Kushner, "Jump diffusion approximations for ordinary differential equations with random right hand sides", submitted to SIAM J. on Control; see also LCDS Report 78-1, September 1978, Brown University.Google Scholar
  8. [8]
    T. G. Kurtz, "Semigroups of conditional shifts and approximation of Markov processes", Ann. Prob., (1975), 4, 618–642.Google Scholar
  9. [9]
    IEEE Tras. on Antennas and Propagation, (1976), AP-24. Special Issue on Adaptive Antenna Arrays.Google Scholar
  10. [10]
    L. E. Brennan, E. L. Pugh and I. S. Reed, "Control-loop noise in adaptive antenna arrays", IEEE Trans. on Aerospace and Electronic Systems, (1971), AES-7, 254–262.Google Scholar
  11. [11]
    I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, (1965), Saunders, Philadelphia.Google Scholar
  12. [12]
    P. Billingsley, Convergence of Porbability Measures, (1968), John Wiley and Sons, New York.Google Scholar
  13. [13]
    G. L. Blankenship and G. C. Papanicolaou, "Stability and control of stochastic systems with wide-band noise disturbances", (1978), preprint.Google Scholar
  14. [14]
    H. J. Kushner, Probability Methods for Approximations for Elliptic Equations and Optimal Stochastic Control Problems, Academic Press, New York, 1977.Google Scholar
  15. [15]
    E. F. Infante, "On the stability of some linear autonomous random systems", (1968), ASME J. Appl. Mech., 35, 7–12.Google Scholar
  16. [16]
    F. Kozin and C. M. Wu, "On the stability of linear stochastic differential equations", (1973), ASME J. Appl. Mech., 40, 87–92.Google Scholar
  17. [17]
    G. Blankenship, "Stability of linear differential equations with random coefficients", (1977), IEEE Trans. on Automatic Control, AC-22, 834–838.Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Harold J. Kushner
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidence

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