## Continuous exponential martingales and BMO.(English)Zbl 0806.60033

Lecture Notes in Mathematics. 1579. Berlin: Springer-Verlag. vii, 91 p. DM 34.00; öS 265.20; sFr 34.00/pbk (1994).
This book deals with continuous local martingales and with the BMO space which is defined by the norm $\sup_ T \biggl \| E \bigl[ | M_ \infty - M_ T | \mid {\mathcal F}_ T \bigr] \biggr \|_ \infty < \infty,$ where the supremum is taken over all stopping times $$T$$. Chapter 1 contains some properties of an exponential local martingale $${\mathcal E} (M): = \exp (M - {1 \over 2} \langle M \rangle)$$, where $$\langle M \rangle$$ denotes the quadratic variation of the martingale $$M$$. In Chapter 2 the John-Nirenberg inequality and the duality between the Hardy space $$H_ 1$$ and BMO is proved. Moreover, it is verified that the following three conditions are equivalent:
(i) $$M \in \text{BMO}$$;
(ii) $${\mathcal E} (M)$$ is a uniformly integrable martingale which satisfies the reverse Hölder inequality: $$E[{\mathcal E} (M)^ p_ \infty \mid {\mathcal F}_ T] \leq C_ p {\mathcal E} (M)^ p_ T$$ for some $$p > 1$$, where $$T$$ is an arbitrary stopping time;
(iii) $${\mathcal E} (M)$$ satisfies the condition $$\sup_ T \| E [\{{\mathcal E} (M)_ T/{\mathcal E} (M)_ \infty \}^{1/(p - 1)} \mid {\mathcal F}_ T \|_ \infty < \infty$$ for some $$p > 1$$.
The distances between $$L_ \infty$$ and BMO and between $$H_ \infty$$ and BMO are considered. In the last chapter we can find some interesting connections between the BMO closure of $$L_ \infty$$ and condition (ii) and between the BMO closure of $$H_ \infty$$ and condition (iii). Finally, some weighted norm and ratio inequalities are proved. There are several counterexamples in the book.
Reviewer: F.Weisz (Budapest)

### MSC:

 60G44 Martingales with continuous parameter 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 42B30 $$H^p$$-spaces
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