Sharp moment inequalities for differentially subordinated martingales

Tom 201 / 2010

Adam Os/ekowski Studia Mathematica 201 (2010), 103-131 MSC: Primary 60G42; Secondary 60G44. DOI: 10.4064/sm201-2-1

Streszczenie

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 ${\mit\Phi}$.

Autorzy

  • Adam Os/ekowskiDepartment of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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