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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sharp maximal inequality for stochastic integrals


Author: Adam Osekowski
Journal: Proc. Amer. Math. Soc. 136 (2008), 2951-2958
MSC (2000): Primary 60HO5; Secondary 60G42
Published electronically: April 14, 2008
MathSciNet review: 2399063
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X=(X_t)_{t\geq 0}$ be a nonnegative supermartingale and $ H=(H_t)_{t\geq 0}$ be a predictable process with values in $ [-1,1]$. Let $ Y$ denote the stochastic integral of $ H$ with respect to $ X$. The paper contains the proof of the sharp inequality

$\displaystyle \sup_{t\geq 0}\vert\vert Y_t\vert\vert _1 \leq \beta_0 \vert\vert\sup_{t\geq 0}X_t\vert\vert _1,$

where $ \beta_0=2+(3e)^{-1}=2,1226\ldots$. A discrete-time version of this inequality is also established.


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Additional Information

Adam Osekowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: ados@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09305-2
PII: S 0002-9939(08)09305-2
Keywords: Martingale, supermartingale, martingale transform, norm inequality, stochastic integral, maximal inequality
Received by editor(s): June 21, 2007
Published electronically: April 14, 2008
Additional Notes: The author was supported by MEiN Grant 1 PO3A 012 29
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.