Let X̂=C−Y where Y is a general one-dimensional Lévy process and C an independent subordinator. Consider the times when a new supremum of X̂ is reached by a jump of the subordinator C. We give a necessary and sufficient condition in order for such times to be discrete. When this is the case and X̂ drifts to −∞, we decompose the absolute supremum of X̂ at these times, and derive a Pollaczek–Hinchin-type formula for the distribution function of the supremum.
@article{1222261921,
author = {Song, Renming and Vondra\v cek, Zoran},
title = {On suprema of L\'evy processes and application in risk theory},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {44},
number = {2},
year = {2008},
pages = { 977-986},
language = {en},
url = {http://dml.mathdoc.fr/item/1222261921}
}
Song, Renming; Vondraček, Zoran. On suprema of Lévy processes and application in risk theory. Ann. Inst. H. Poincaré Probab. Statist., Tome 44 (2008) no. 2, pp. 977-986. http://gdmltest.u-ga.fr/item/1222261921/