Translator Disclaimer
2013 Explicit formula for the supremum distribution of a spectrally negative stable process
Zbigniew Michna
Author Affiliations +
Electron. Commun. Probab. 18: 1-6 (2013). DOI: 10.1214/ECP.v18-2236

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.

 

Citation

Download Citation

Zbigniew Michna. "Explicit formula for the supremum distribution of a spectrally negative stable process." Electron. Commun. Probab. 18 1 - 6, 2013. https://doi.org/10.1214/ECP.v18-2236

Information

Accepted: 2 February 2013; Published: 2013
First available in Project Euclid: 7 June 2016

zbMATH: 1323.60064
MathSciNet: MR3033593
Digital Object Identifier: 10.1214/ECP.v18-2236

Subjects:
Primary: 60G51
Secondary: 60G52, 60G70

JOURNAL ARTICLE
6 PAGES


SHARE
Back to Top