Probability Theory and Related Fields

, Volume 78, Issue 1, pp 113–126 | Cite as

A new aspect ofL in the space of BMO-martingales

  • Norihiko Kazamaki
Article

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.

Keywords

Real Number Stochastic Process Probability Theory Mathematical Biology Related Problem 

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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Norihiko Kazamaki
    • 1
  1. 1.Department of MathematicsToyama UniversityToyamaJapan

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