Let X=(Xt)t≥0 be a stable Lévy process of index α∈(1, 2) with no negative jumps and let St=sup0≤s≤t Xs 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.
Publié le : 2008-09-15
Classification:
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,
60G52,
45D05,
60J75,
45E99,
26A33
@article{1221138766,
author = {Bernyk, Violetta and Dalang, Robert C. and Peskir, Goran},
title = {The law of the supremum of a stable L\'evy process with no negative jumps},
journal = {Ann. Probab.},
volume = {36},
number = {1},
year = {2008},
pages = { 1777-1789},
language = {en},
url = {http://dml.mathdoc.fr/item/1221138766}
}
Bernyk, Violetta; Dalang, Robert C.; Peskir, Goran. The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab., Tome 36 (2008) no. 1, pp. 1777-1789. http://gdmltest.u-ga.fr/item/1221138766/