## 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

Zbl 0806.60033
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