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A characterisation of the closure of H in BMO

  • W. Sciiachermayer
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1626)

Abstract

We show that a continuous martingale M∈BMO has a ‖·‖BMO2 distance to H less than ε>0 iff M may be written as a finite sum \(M = \sum\limits_{n = 0}^N {{}^{T_n }M^{T_{n + 1} } } \) such that, for each 0≤nN, we have \(\parallel {}^{T_n }M^{T_{n + 1} } \parallel _{BMO_2 } < \varepsilon \). In particular, we obtain a characterisation of the BMO-closure of H.

This result was motivated by some problems posed in the survey of N. Kazamaki [K

1980 Mathematics Subject Classification (1991 Revision)

Primary 60 G 48 60 H 05 Key words and phrases Martingales Bounded Mean Oscillation H space 

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References

  1. [DMSSS 94] F. Delbaen, P. Monat, W. Schachermayer, M. Schweizer, C. Stricker, Inégalités de normes avec Poids et Fermeture d’un Espace d’Intégrales Stochastiques, CRAS, Paris 319, Série I (1994), 1079–1081.Google Scholar
  2. [DMSSS 95]. F. Delbaen, P. Monat, W. Schachermayer, M. Schweizer, C. Stricker, Weighted Norm Inequalities and Closedness of a Space of Stochastic Integrals, preprint, (1995), 45p.Google Scholar
  3. [DM 79]. C. Doléans-Dade, P.A. Meyer, Inégalités de normes avec poids, Séminaire de Probabilités XIII, Springer Lecture Notes in Mathematics 721 (1979), 313–331.CrossRefzbMATHGoogle Scholar
  4. [K 94]. N. Kazamaki, Continuous Exponential Martingales and BMO, Springer Lecture Notes in Mathematics 1579 (1994).Google Scholar
  5. [RY 91]. D. Revuz, M. Yor, Continuous Martingales and Browian Motion, Springer, Berlin-Heidelberg-New York, (1991).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • W. Sciiachermayer
    • 1
  1. 1.Institut für Statistik der Universität WienWienAustria

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