2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/138047Let Z be a strictly a-stable real Levy process (a>1) and X be a fluctuating b-homogeneous additive functional of Z. We investigate the asymptotics of the first passage-time of X above 1, and give a general upper bound. When Z has no negative jumps, we prove that this bound is optimal and does not depend on the homogeneity parameter b. This extends a result of Y. Isozaki and solves partially a conjecture of Z. Shi.Revised version. To appear in ALEA Latin American Journal of Probability and Mathematical StatisticsProbability60F99, 60G18, 60G52, 60J55text