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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: 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

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.

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