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September 2008 The law of the supremum of a stable Lévy process with no negative jumps
Violetta Bernyk, Robert C. Dalang, Goran Peskir
Ann. Probab. 36(5): 1777-1789 (September 2008). DOI: 10.1214/07-AOP376

Abstract

Let X=(Xt)t≥0 be a stable Lévy process of index α∈(1, 2) with no negative jumps and let St=sup0≤stXs denote its running supremum for t>0. We show that the density function ft of St can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann–Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for ft. Recalling the familiar relation between St and the first entry time τx of X into [x, ∞), this further translates into an explicit series representation for the density function of τx.

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Violetta Bernyk. Robert C. Dalang. Goran Peskir. "The law of the supremum of a stable Lévy process with no negative jumps." Ann. Probab. 36 (5) 1777 - 1789, September 2008. https://doi.org/10.1214/07-AOP376

Information

Published: September 2008
First available in Project Euclid: 11 September 2008

zbMATH: 1185.60051
MathSciNet: MR2440923
Digital Object Identifier: 10.1214/07-AOP376

Subjects:
Primary: 45D05, 60G52
Secondary: 26A33, 45E99, 60J75

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 5 • September 2008
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