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Nonnegative Supermartingales and Martingales, and the Girsanov Theorem

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

Abstract

Let (Ω, F, P) be a complete probability space, and let (F t ), 0 ≤ t T, be a nondecreasing family of sub-σ-algebras of F, augmented by sets from F of probability zero. Let W = (W t , F t ) be a Wiener process and let γ = (γ t , F t ) be a random process with
$$P\left( {\int {_0^T} \gamma _s^2ds \infty } \right) = 1.$$
(6.1)

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Bibliography

Notes and References. 1

  1. 248.
    Novikov, A.A. (1972): On an identity for stochastic integrals. Teor. Veroyatn. Primen., 17, 4, 761 - 5Google Scholar
  2. 75.
    Gikhman, I.I. and Skorokhod, A.V. (1980): Stochastic Differential Equations. Springer-Verlag, Berlin Heidelberg New YorkGoogle Scholar
  3. 76.
    Girsanov, I.V. (1960): On transformation of one class of random processes with the help of absolutely continuous change of the measure. Teor. Veroyatn. Primen., 5, 1, 314 - 30MathSciNetGoogle Scholar
  4. 212.
    Liptser, R.S. and Shiryaev, A.N. (1972): On absolute continuity of measures corresponding to diffusion type processes with respect to a Wiener measure. Izv. Akad. Nauk SSSR, Ser. Mat., 36, 4, 874 - 89Google 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. 288.
    Shiryaev, A.N. (1999): Essentials of Stochastic Finance. World Scientific, SingaporezbMATHGoogle Scholar
  3. 106.
    Jacod, J. and Shiryaev, A.N. (1987): Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin Heidelberg New YorkzbMATHCrossRefGoogle Scholar
  4. 214.
    Liptser, R.S. and Shiryaev, A.N. (1989): Theory of Martingales. Kluwer, Dordrecht (Russian edition 1986 )zbMATHCrossRefGoogle 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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