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A characterisation of the closure of \(H^ \infty\) in BMO. (English) Zbl 0856.60046

Azéma, J. (ed.) et al., Séminaire de probabilités XXX. Berlin: Springer. Lect. Notes Math. 1626, 344-356 (1996).
Summary: We show that a continuous martingale \(M\in \text{BMO}\) has a \(|\cdot|_{\text{BMO}_2}\)-distance to \(H^\infty\) less than \(\varepsilon> 0\) iff \(M\) may be written as a finite sum \(M= \sum^N_{n= 0} {^{T_n} M^{T_{n+ 1}}}\) such that, for each \(0\leq n\leq N\), we have \(|^{T_n} M^{T_{n+ 1}}|_{\text{BMO}_2}< \varepsilon\). In particular, we obtain a characterization of the BMO-closure of \(H^\infty\). This result was motivated by some problems posed in the survey of N. Kazamaki [“Continuous exponential martingales and BMO” (1994; Zbl 0806.60033)]. We also give answers to some other questions, pertaining to BMO-martingales, which have been raised by N. Kazamaki (loc. cit.).
For the entire collection see [Zbl 0840.00041].

MSC:

60G48 Generalizations of martingales
60H05 Stochastic integrals

Citations:

Zbl 0806.60033
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