Square Integrable Martingales and Structure of the Functionals on a Wiener Process

  • Robert S. Liptser
  • Albert N. Shiryaev
Part of the Applications of Mathematics book series (SMAP, volume 5)


Let (Ω, F, P) be a complete probability space, and let F = (F t ),t ≥ 0, be a nondecreasing (right continuous) family of sub-σ-algebras of F, each of which is augmented by sets from F having zero P-probability.


Continuous Modification Simple Function Wiener Process Diffusion Type Stochastic Integral 
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Notes and References. 1

  1. 4.
    Anulova, A., Veretennikov, A., Krylov, N., Liptser, R.S. and Shiryaev, A.N. (1998): Stochastic Calculus. Encyclopedia of Mathematical Sciences. vol 45, Springer-Verlag, Berlin Heidelberg New YorkGoogle Scholar
  2. 40.
    Clark, I.M. (1970): The representation of functionals of Brownian motion by stochastic integrals. Ann. Math. Stat., 41, 4, 1282 - 95zbMATHCrossRefGoogle Scholar
  3. 41.
    Courrège, Ph. (1962-3): Intégrales stochastiques et martingales de carré intégrable. Sémin. Brelot—Choquet—Deny (7th year)Google Scholar
  4. 66.
    Fujisaki, M., Kallianpur, G. and Kunita, H. (1972): Stochastic differential equations for the nonlinear filtering problem. Osaka J. Math., 9, 1, 19 - 40MathSciNetzbMATHGoogle Scholar
  5. 106.
    Jacod, J. and Shiryaev, A.N. (1987): Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin Heidelberg New YorkzbMATHCrossRefGoogle Scholar

Notes and References.2

  1. 142.
    Karatzas, I. and Shreve, S.E. (1991): Brownian Motion and Stochastic Calculus. Springer-Verlag, New York Berl in HeidelbergzbMATHGoogle Scholar
  2. 137.
    Kallianpur, G. and Striebel, C. (1969): Stochastic differential equations occurring in the estimation of continuous parameter stochastic processes. Teor. Veroyatn. Primen., 14, 4, 597 - 622MathSciNetzbMATHGoogle Scholar
  3. 171.
    Kunita, H. and Watanabe, Sh. (1967): On square integrable martingales. Nagoya Math. J., 30, 209 - 45MathSciNetzbMATHGoogle Scholar
  4. 214.
    Liptser, R.S. and Shiryaev, A.N. (1989): Theory of Martingales. Kluwer, Dordrecht (Russian edition 1986 )zbMATHCrossRefGoogle Scholar
  5. 229.
    Meyer, P.A. (1966): Probabilités et Potentiel. Hermann, PariszbMATHGoogle Scholar
  6. 261.
    Revuz, D. and Yor, M. ( 1991, 1994, 1998 ): Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin Heidelberg New YorkGoogle Scholar
  7. 303.
    Wentzell, A.D. (1961): Additive functionals of a multivariate Wiener process. Dokl. Akad. Nauk SSSR, 130, 1, 13 - 6Google Scholar
  8. 326.
    Yershov, M.P. (1970): Sequential estimation of diffusion processes. Teor. Veroyatn. Primen., 15, 4, 705 - 17Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Robert S. Liptser
    • 1
  • Albert N. Shiryaev
    • 2
  1. 1.Department of Electrical Engineering SystemsTel Aviv UniversityTel AvivIsrael
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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