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Maximal inequalities for continuous martingales and their differential subordinates

Author: Adam Osȩkowski
Journal: Proc. Amer. Math. Soc. 139 (2011), 721-734
MSC (2010): Primary 60G44; Secondary 60H05
Published electronically: August 13, 2010
MathSciNet review: 2736351
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Abstract: Let $ X=(X_t)_{t\geq 0}, Y=(Y_t)_{t\geq 0}$ be continuous-path martingales such that $ Y$ is differentially subordinate to $ X$. The paper contains the proofs of the sharp inequalities

$\displaystyle \sup_{t\geq 0}\vert\vert Y_t\vert\vert _p \leq \sqrt{\frac{2}{p}} \vert\vert\sup_{t\geq 0}\vert X_t\vert \vert\vert _p, \quad 1\leq p< 2$


$\displaystyle \sup_{t\geq 0}\vert\vert Y_t\vert\vert _p \leq (p-1) \vert\vert\sup_{t\geq 0}\vert X_t\vert \vert\vert _p, \quad 2\leq p<\infty.$

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Additional Information

Adam Osȩkowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Keywords: Martingale, stochastic integral, maximal inequality, differential subordination
Received by editor(s): June 29, 2009
Received by editor(s) in revised form: April 14, 2010
Published electronically: August 13, 2010
Additional Notes: The author was partially supported by the Foundation for Polish Science and MNiSW Grant N N201 397437
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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