On the distance between $\langle X \rangle$ and $L^{\infty} $ in the space of continuous BMO-martingales

Litan Yan ¹   ; Norihiko Kazamaki ²

¹ Department of Mathematics College of Science Donghua University 1882 West Yan'an Rd. Shanghai 200051, P.R. China
² Department of Mathematics Toyama University 3190 Gofuku,Toyama 930-8555, Japan

Studia Mathematica, Tome 168 (2005) no. 2, pp. 129-134

Résumé

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 $X$.

Détail

DOI : 10.4064/sm168-2-3

Keywords: mathcal continuous bmo martingale bmo equiv sup infty x mathcal infty infty where supremum taken stopping times nbsp define critical exponent nbsp sup exp langle rangle infty langle rangle mathcal infty infty where supremum taken stopping times consider continuous martingale defined langle rangle infty mathcal e langle rangle infty mathcal characterize distance between langle rangle class infty bounded martingales space continuous bmo martingales inequalities frac infty leq leq frac infty every continuous bmo martingale