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