The Annals of Probability

On Wiener-Hopf Factorisation and the Distribution of Extrema for Certain Stable Processes

R. A. Doney

Full-text: Open access

Abstract

It is shown that when the index $0 < \alpha < 2, \alpha \neq 1$, and the symmetry parameter $-1 \leq \beta \leq 1$ of a stable process $\{X(t); t \geq 0\}$ are such that $P\{X(1) > 0\} = l\alpha^{-1} - k$, where $l$ and $k$ are integers, Darling's integral can be evaluated. This leads to explicit formulas for a transform of the Laplace transform of $\sup_{0\leq t\leq 1}X(t)$ and the Wiener-Hopf factors of $\{X(t), t \geq 0\}$.

Article information

Source
Ann. Probab., Volume 15, Number 4 (1987), 1352-1362.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991981

Digital Object Identifier
doi:10.1214/aop/1176991981

Mathematical Reviews number (MathSciNet)
MR905336

Zentralblatt MATH identifier
0631.60069

JSTOR
links.jstor.org

Subjects
Primary: 60J30

Keywords
Stable processes Wiener-Hopf factorisation supremum functional Levy process Darling's integral

Citation

Doney, R. A. On Wiener-Hopf Factorisation and the Distribution of Extrema for Certain Stable Processes. Ann. Probab. 15 (1987), no. 4, 1352--1362. doi:10.1214/aop/1176991981. https://projecteuclid.org/euclid.aop/1176991981


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