## Electronic Communications in Probability

### Maximal weak-type inequality for stochastic integrals

Adam Osekowski

#### 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

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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