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Martingales

  • Daniel Revuz
  • Marc Yor
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 293)

Abstract

Martingales are a very important subject in their own right as well as by their relationship with analysis. Their kinship to BM will make them one of our main subjects of interest as well as one of our foremost tools. In this chapter, we describe some of their basic properties which we shall use throughout the book.

Keywords

Brownian Motion Iterate Logarithm Maximal Inequality Regularization Theorem Strong Markov Property 
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Notes and Comments

  1. 1.
    Doob, J.L. Stochastic processes. Wiley, New York 1953zbMATHGoogle Scholar
  2. 1.
    Dellacherie, C. Intégrales stochastiques par rapport aux processus de Wiener et de Poisson. Sém. Prob. VIII. Lecture Notes in Mathematics, vol. 381. Springer, Berlin Heidelberg New York 1974, pp. 25 - 26Google Scholar
  3. 1.
    Ikeda, N. and Manabe, S. ntegral of differential forms along the paths of diffusion processes. Publ. R.I.M.S., Kyoto Univ. 15 (1978) 827 - 852Google Scholar
  4. 1.
    Khintchine, A.Y. Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. (Ergebnisse der Mathematik, Bd. 2). Springer Berlin Heidelberg 1933, pp. 72 - 75Google Scholar
  5. 1.
    Kolmogorov, A.N. Uber das Gesetz das itierten Logarithmus. Math. Ann. 101 (1929) 126 - 135MathSciNetCrossRefGoogle Scholar
  6. 10.
    Yor, M. Loi de l’indice du lacet brownien et distribution de Hartman-Watson. Z.W. 53 (1980) 71 - 95MathSciNetzbMATHCrossRefGoogle Scholar
  7. 1.
    Itô, K., and McKean, H.P.Diffusion processes and their sample paths. Springer, Berlin Heidelberg New York 1965CrossRefGoogle Scholar
  8. 1.
    Meyer, P.A., Smythe, R.T., and Walsh, J.B. Birth and death of Markov processes. Proc. Sixth Berkeley Symposium III. University of California Press, 1971, pp. 295 - 305Google Scholar
  9. 1.
    Orey, S. Two strong laws for shrinking Brownian tubes. Z.W. 63 (1983) 281 - 288MathSciNetzbMATHCrossRefGoogle Scholar
  10. 1.
    Neveu, J. Processus aléatoires gaussiens. Les Presses de l’Univ. de Montréal, 1968Google Scholar
  11. 1.
    Dubins, L. ises and uperossings of non-negative martingales. Ill. J. Math. 6 (1962) 226 - 241MathSciNetzbMATHGoogle Scholar
  12. 1.
    Bernard, A., and Maisonneuve, B. Décomposition atomique de martingales de la classe H 1. Sém. Prob. XI. Lecture Notes in Mathematics, vol. 581. Springer, Berlin Heidelberg New York 1977, pp. 303 - 323Google Scholar
  13. 1.
    Lenglart, E., Lépingle, D., and Pratelli, M. Une présentation unifiée des inégalités en théorie des martingales. Sém. Prob. XIV. Lecture Notes in Mathematics, vol. 784. Springer, Berlin Heidelberg New York 1980, pp. 26 - 48Google Scholar
  14. 1.
    Dellacherie, C. Intégrales stochastiques par rapport aux processus de Wiener et de Poisson. Sém. Prob. VIII. Lecture Notes in Mathematics, vol. 381. Springer, Berlin Heidelberg New York 1974, pp. 25 - 26Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Daniel Revuz
    • 1
  • Marc Yor
    • 2
  1. 1.Département de MathématiquesUniversité de Paris VIIParis Cedex 05France
  2. 2.Laboratoire de ProbabilitésUniversité Pierre et Marie CurieParis Cedex 05France

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