The Annals of Probability

The law of the supremum of a stable Lévy process with no negative jumps

Violetta Bernyk, Robert C. Dalang, and Goran Peskir

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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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Ann. Probab. Volume 36, Number 5 (2008), 1777-1789.

First available in Project Euclid: 11 September 2008

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Zentralblatt MATH identifier

Primary: 60G52: Stable processes 45D05: Volterra integral equations [See also 34A12]
Secondary: 60J75: Jump processes 45E99: None of the above, but in this section 26A33: Fractional derivatives and integrals

Stable Lévy process with no negative jumps spectrally positive running supremum process first hitting time first entry time weakly singular Volterra integral equation polar kernel Riemann–Liouville fractional differential equation Abel equation Wiener–Hopf factorization


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

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