Sharp Norm Inequalities for Martingales and their Differential Subordinates

Volume 55 / 2007

Adam Os/ekowski Bulletin Polish Acad. Sci. Math. 55 (2007), 373-385 MSC: Partially supported by MEiN Grant 1 PO3A 012 29. DOI: 10.4064/ba55-4-9

Abstract

Suppose $f=(f_n)$, $g=(g_n)$ are martingales with respect to the same filtration, satisfying $$ |f_n-f_{n-1}| \leq |g_n-g_{n-1}|,\quad \ n=1,2,\ldots, $$ with probability $1$. Under some assumptions on $f_0$, $g_0$ and an additional condition that one of the processes is nonnegative, some sharp inequalities between the $p$th norms of $f$ and $g$, $0< p< \infty$, are established. As an application, related sharp inequalities for stochastic integrals and harmonic functions are obtained.

Authors

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

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