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)


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