Two remarks on the class of bounded continuous martingales (Q2758288)

It is known, see \textit{N. Kazamaki} [``Continuous exponential martingales and BMO'' (1994; Zbl 0806.60033)], that a continuous martingale \(M\) belongs to the BMO space if and only if the exponential martingale \({\mathcal E} (M): = \exp (M - {1 \over 2} \langle M \rangle)\) satisfies the \((A_p)\) condition for some \(p>1\), that is NEWLINE\[NEWLINE \sup_T \bigl\|E \bigl[\bigl\{{\mathcal E} (M)_T/{\mathcal E} (M)_\infty \bigr\}^{1/(p - 1)} \mid {\mathcal F}_T \bigr]\bigr\|_\infty < \infty, NEWLINE\]NEWLINE where the supremum is taken over all stopping times \(T\). It is shown that if \(M\) belongs to the BMO-closure of \(L^\infty\), then \({\mathcal E} (M)\) satisfies \((A_p)\) if and only if \(1/\sqrt{2(p-1)}< b(M)\), where NEWLINE\[NEWLINE b(M)=\sup\left\{\beta>0: \sup_T \|E[\exp(\beta^2(\langle M \rangle_\infty - \langle M \rangle_T)) \mid {\mathcal F}_T]\|_\infty < \infty \right\}.NEWLINE\]