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Sharp maximal inequality for martingales and stochastic integrals


 
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1. Title Title of document Sharp maximal inequality for martingales and stochastic integrals
 
2. Creator Author's name, affiliation, country Adam Osekowski; University of Warsaw
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Martingale; stochastic integral; maximal function
 
3. Subject Subject classification 60G42; 60G44
 
4. Description Abstract Let $X=(X_t)_{t\geq 0}$ be a martingale and $H=(H_t)_{t\geq 0}$ be a predictable process taking values in $[-1,1]$. Let $Y$ denote the stochastic integral of $H$ with respect to $X$. We show that $$ ||\sup_{t\geq 0}Y_t||_1 \leq \beta_0 ||\sup_{t\geq 0}|X_t|||_1,$$ where $\beta_0=2,0856\ldots$ is the best possible. Furthermore, if, in addition, $X$ is nonnegative, then $$ ||\sup_{t\geq 0}Y_t||_1 \leq \beta_0^+ ||\sup_{t\geq 0}X_t||_1,$$ where $\beta_0^+=\frac{14}{9}$ is the best possible.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Partially supported by MEiN Grant 1 PO3A 012 29 and Foundation for Polish Science
 
7. Date (YYYY-MM-DD) 2009-01-23
 
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/1438
 
10. Identifier Digital Object Identifier 10.1214/ECP.v14-1438
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 14
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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