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Maximal weak-type inequality for stochastic integrals


 
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1. Title Title of document Maximal weak-type inequality for stochastic integrals
 
2. Creator Author's name, affiliation, country Adam Osekowski; University of Warsaw; Poland
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Martingale;maximal;weak type inequality;best constant
 
3. Subject Subject classification 60G44;60G42
 
4. Description 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 $$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 2\mathbb{E} \sup_{t\geq 0}X_t.$$
(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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) NCN grant DEC-2012/05/B/ST1/00412
 
7. Date (YYYY-MM-DD) 2014-05-04
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/3151
 
10. Identifier Digital Object Identifier 10.1214/ECP.v19-3151
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 19
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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