On Strict-Sense Forms of the Hida-Cramer Representation

  • Frank B. Knight
Part of the Progress in Probability and Statistics book series (PRPR, volume 9)


This paper is an attempt to delineate more clearly than before the role of the standard Brownian motion and Poisson processes in generating general stochastic processes (Ω δt,Xt, P). In discrete time, the analog would be to use a sequence of independent Bernoulli random walks. This is a very different setting, and one about which we have nothing to contribute. Evidently such a sequence does not go far toward generating a general discrete parameter process, at least in the sense we have in mind here. The situation in continuous time, however, is antithetical. One can obtain representation theorems of considerable generality, which to some extent crystalize an important aspect of all continuous time processes. We consider this as the aspect which involves randomness of time without randomness of place. In some sense, there is a natural dichotomy of the two kinds of randomness, and our aim is to isolate and study the case in which the role of randomness of place can be eliminated. A basic tool in our investigation is the “prediction process” Zt of [11], but it is no more than that. Theoretically, one could attempt to define respresentations of Zt itself, analogous to those obtained below but valid for all P. In the present paper, however, each P is treated separately.


Brownian Motion Poisson Process Wiener Process Gaussian Case Continuous Component 


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  1. 1.
    H. CRAMER. On some classes of non-stationary stochastic processes. Proa, of the Fourth Berkeley Symposium II, J. Neyman, Ed. Univ. of Cal. Press, 1961, pp. 57–78.Google Scholar
  2. 2.
    H. CRAMER. Stochastic processes as curves in Hilbert space.Theory of Probability and Its Applications, IX (1964), 169–177.CrossRefGoogle Scholar
  3. 3.
    M.H.A. DAVIS and P. VARAIYA. The multiplicity of an increasing family of σ-fields. The Annals of Probability 2 (1974), 958–963.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    C. DELLACHERIE and P.A. MEYER. Probabilités et Potentiel. Pubi, de Lfinstitut de Math, de LfUniv. Strasbourg No. XV(1975) and No. XVII (1980).Google Scholar
  5. 5.
    J.L. DOOB. Stochastic Processes. Wiley, New York, 1953.MATHGoogle Scholar
  6. 6.
    B.V. GNEDENKO and A.N. K0LM0G0R0V. Limit distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, 1954.MATHGoogle Scholar
  7. 7.
    R.K. Getoor. Markov Processes: Ray Processes and Right Processes. Lecture Notes in Math. No. 440. Springer-Verlag, Berlin, 1975.MATHGoogle Scholar
  8. 8.
    S.W. He and J.G. Wang. The total continuity of natural filtrations. Sern, de Prob. XVI, pp. 348–354. Lecture Notes in Math. No. 920. Springer-Verlag, Berlin, 1981.Google Scholar
  9. 9.
    T. Hida. Canonical representations of Gaussian processes and their applications. Memoirs of the College of Science, Univ. of Kyoto Series A, (1), Vol. 33 (1960), 109–155.MathSciNetMATHGoogle Scholar
  10. 10.
    F.B. KNIGHT. A reduction of continuous square-integrable martingales to Brownian motion. Lecture Notes in Math. No. 190. Springer-Verlag, Berlin, 1971.Google Scholar
  11. 11.
    F.B. Knight. Essays on the Prediction Process. Inst, of Math. Statistics Lecture Notes Series No. 1¿ 1981.Google Scholar
  12. 12.
    F.B. Knight. A post-predictive view of Gaussian processes. Ann. Sci. de L’Êcole Normale Supérieure, Series 4, Vol. 16 (1983), 541–566.MathSciNetGoogle Scholar
  13. 13.
    H. Kunita and S. Watanabe. On square-integrable martingales. Nagoya Math. J. 30 (1967), 209–245.MathSciNetMATHGoogle Scholar
  14. 14.
    J. de SAM LAZARO and P.A. MEYER. Questions de la théorie des flots VI. Sem. de Prob. IX, pp. 73–88. Lecture Notes in Math. No. 465. Springer-Verlag, Berlin, 1981.Google Scholar
  15. 15.
    Y. Le Jan. Temps dfarrêt stricts et martingales de saut. Z. Wahrscheinlichkeitstheorie verw. Geb. 44 (1978), 213–225.MATHCrossRefGoogle Scholar
  16. 16.
    Y. Le Jan. Martingales et changement de temps. Sem. de Prob. XIII, pp. 385–400. Lecture Notes in Math. No. 721. Springer-Verlag, Berlin, 1979.Google Scholar
  17. 17.
    P.A. Meyer. Intégrales stochastiques III.Sem. de Prob. I. Lecture Notes in Math. No. 39. Springer-Verlag, Berlin, 1967.Google Scholar
  18. 18.
    P.A. Meyer. Demonstration simplifiée d’un théorème de Knight. Sem. de Prob. V, pp. 191–195. Lecture Notes in Math. No. 191. Springer-Verlag, Berlin, 1971.Google Scholar
  19. 19.
    S. Watanabe. On discontinuous additive functionals and Lévy measures of a Markov process. Jap. J. Math. 34 (1964), 53–79.MATHGoogle Scholar
  20. 20.
    J. Walsh and P.A. Meyer. Quelques applications des résolvantes de Ray. Invent. Math. 14 (1971), 143–166.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston, Inc. 1986

Authors and Affiliations

  • Frank B. Knight
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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