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A new aspect ofL ∞ in the space of BMO-martingales
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  • Published: March 1988

A new aspect ofL ∞ in the space of BMO-martingales

  • Norihiko Kazamaki1 

Probability Theory and Related Fields volume 78, pages 113–126 (1988)Cite this article

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Summary

Let\(M = \left( {M_t , \mathfrak{F}_t } \right)\) be a continuous BMO-martingale. Then the associated exponential martingale ℰ(M) satisfies the reverse Hölder inequality

$$\left( {R_p } \right)E[\mathcal{E}\left( M \right)_\infty ^p |\mathfrak{F}_T ] \leqq C_p \mathcal{E}\left( M \right)_T^p $$

for somep>1, whereT is an arbitrary stopping time (see [3, 5]). Our claim is, in a word, that the (R p ) condition bears upon the distance betweenM andL ∞ if BMO. Especially, we shall prove thatM belongs to the BMO-closure ofL ∞ if and only if ℰ(λM) satisfies all (R p ) for every real number λ. Some related problems are also considered.

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References

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Authors and Affiliations

  1. Department of Mathematics, Toyama University, Gofuku, 930, Toyama, Japan

    Norihiko Kazamaki

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  1. Norihiko Kazamaki
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Kazamaki, N. A new aspect ofL ∞ in the space of BMO-martingales. Probab. Th. Rel. Fields 78, 113–126 (1988). https://doi.org/10.1007/BF00718039

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  • Received: 20 December 1986

  • Revised: 20 November 1987

  • Issue Date: March 1988

  • DOI: https://doi.org/10.1007/BF00718039

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Keywords

  • Real Number
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Related Problem
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