On the distance between $\langle X \rangle$ and $L^{\infty} $ in the space of continuous BMO-martingales
Volume 168 / 2005
Abstract
Let $X=(X_t,{\mathcal F}_t)$ be a continuous BMO-martingale, that is, $$ \|X\|_{\rm BMO}\equiv \sup_T\|E[|X_\infty-X_T|\,|\,{\mathcal F}_T]\|_\infty<\infty, $$ where the supremum is taken over all stopping times $T$. Define the critical exponent $b(X)$ by $$ b(X)=\{b>0:\sup_T\|E[\exp(b^2(\langle X \rangle_\infty-\langle X \rangle_T))\,|\,{\mathcal F}_T]\|_\infty<\infty\}, $$ where the supremum is taken over all stopping times $T$. Consider the continuous martingale $q(X)$ defined by $$ q(X)_t=E[\langle X \rangle_\infty\,|\,{\mathcal F}_t]-E[\langle X\rangle_\infty\,|\, {\mathcal F}_0]. $$ We use $q(X)$ to characterize the distance between $\langle X \rangle$ and the class $L^\infty$ of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities $$ \frac1{4d_1(q(X),L^\infty)}\leq b(X)\leq \frac4{d_1(q(X),L^\infty)} $$ hold for every continuous BMO-martingale $X$.
