Electronic Communications in Probability

Maximal weak-type inequality for stochastic integrals

Adam Osekowski

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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.

Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 25, 13 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465316727

Digital Object Identifier
doi:10.1214/ECP.v19-3151

Mathematical Reviews number (MathSciNet)
MR3208323

Zentralblatt MATH identifier
1318.60047

Subjects
Primary: 60G44: Martingales with continuous parameter
Secondary: 60G42: Martingales with discrete parameter

Keywords
Martingale maximal weak type inequality best constant

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Osekowski, Adam. Maximal weak-type inequality for stochastic integrals. Electron. Commun. Probab. 19 (2014), paper no. 25, 13 pp. doi:10.1214/ECP.v19-3151. https://projecteuclid.org/euclid.ecp/1465316727


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