Sharp moment inequalities for differentially subordinated martingales

Adam Os/ekowski¹

¹ Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Studia Mathematica, Tome 201 (2010) no. 2, pp. 103-131

Résumé

We determine the optimal constants $C_{p,q}$ in the moment inequalities $$ \|g\|_p \leq C_{p,q} \|f\|_q,\quad\ 1\leq p q \infty, $$ where $f=(f_n)$, $g=(g_n)$ are two martingales, adapted to the same filtration, satisfying $$ |dg_n|\leq |df_n|,\quad\ n=0,1,2,\ldots, $$ with probability $1$. Furthermore, we establish related sharp estimates $$ \|g\|_1 \leq \sup_n {\mathbb E} {\mit\Phi}(|f_n|)+L({\mit\Phi}),$$ where ${\mit\Phi}$ is an increasing convex function satisfying certain growth conditions and $L({\mit\Phi})$ depends only on ${\mit\Phi}$.

Détail

DOI : 10.4064/sm201-2-1

Keywords: determine optimal constants moment inequalities leq quad leq infty where martingales adapted filtration satisfying leq quad ldots probability furthermore establish related sharp estimates leq sup mathbb mit phi mit phi where mit phi increasing convex function satisfying certain growth conditions mit phi depends only mit phi