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Explicit formula for the supremum distribution of a spectrally negative stable process


 
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1. Title Title of document Explicit formula for the supremum distribution of a spectrally negative stable process
 
2. Creator Author's name, affiliation, country Zbigniew Michna; Wroclaw University of Economics; Poland
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Lévy process, distribution of the supremum of a stochastic process, $\alpha$-stable Lévy process
 
3. Subject Subject classification 60G51; 60G52; 60G70
 
4. Description Abstract

In this article we get simple formulas for $E\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative Lévy process with infinite variation. As a consequence we derive a generalization of the well-known formula for the supremum distribution of Wiener process that is we obtain $P(\sup_{s\leq t}Z_{\alpha}(s)\geq u)=\alpha\,P(Z_{\alpha}(t)\geq u)$ for $u\geq 0$ where $Z_{\alpha}$ is a spectrally negative $\alpha$-stable Lévy process with $1<\alpha\leq 2$ which also stems from Kendall's identity for the first crossing time. Our proof uses a formula for the supremum distribution of a spectrally positive Lévy process which follows easily from the elementary Seal's formula.

 
 
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6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2013-02-02
 
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/2236
 
10. Identifier Digital Object Identifier 10.1214/ECP.v18-2236
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 18
 
12. Language English=en en
 
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