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2014 Maximal weak-type inequality for stochastic integrals
Adam Osekowski
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Electron. Commun. Probab. 19: 1-13 (2014). DOI: 10.1214/ECP.v19-3151

Abstract

Assume that $X$ is a real-valued martingale starting from $0$, $H$ is a predictable process with values in $[-1,1]$ and $Y$ is the stochastic integral of $H$ with respect to $X$. The paper contains the proofs of the following sharp weak-type estimates. (i) If $X$ has continuous paths, then $$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 2\mathbb{E} \sup_{t\geq 0}X_t.$$<br />(ii) If $X$ is arbitrary, then$$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 3.477977\ldots\mathbb{E} \sup_{t\geq 0}X_t.$$The proofs rest on Burkholder's method and exploits the existence of certain special functions possessing appropriate concavity and majorization properties.

Citation

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Adam Osekowski. "Maximal weak-type inequality for stochastic integrals." Electron. Commun. Probab. 19 1 - 13, 2014. https://doi.org/10.1214/ECP.v19-3151

Information

Accepted: 4 May 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1318.60047
MathSciNet: MR3208323
Digital Object Identifier: 10.1214/ECP.v19-3151

Subjects:
Primary: 60G44
Secondary: 60G42

JOURNAL ARTICLE
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