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

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dc.creatorBernyk, Violetta
dc.creatorDalang, Robert C.
dc.creatorPeskir, Goran
dc.date2007-06-11
dc.date2008-09-26
dc.date.accessioned2026-07-07T10:05:05Z
dc.date.available2026-07-07T10:05:05Z
dc.descriptionLet $X=(X_t)_{t\ge0}$ be a stable Lévy process of index $α\in(1,2)$ with no negative jumps and let $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t>0$. We show that the density function $f_t$ of $S_t$ 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 $f_t$. Recalling the familiar relation between $S_t$ and the first entry time $τ_x$ of $X$ into $[x,\infty)$, this further translates into an explicit series representation for the density function of $τ_x$.
dc.descriptionPublished in at http://dx.doi.org/10.1214/07-AOP376 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/0706.1503
dc.identifierhttp://arxiv.org/abs/0706.1503
dc.identifierAnnals of Probability 2008, Vol. 36, No. 5, 1777-1789
dc.identifierdoi:10.1214/07-AOP376
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/169907
dc.subjectProbability
dc.subject60G52, 45D05 (Primary) 60J75, 45E99, 26A33 (Secondary)
dc.titleThe law of the supremum of a stable Lévy process with no negative jumps
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