Persistence of integrated stable processes

Abstract

We compute the persistence exponent of the integral of a stable Lévy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Shi (Lower tails of some integrated processes. In: Small deviations and related topics (problem panel 2003). Along the way, we investigate the law of the stable process $$L$$L evaluated at the first time its integral $$X$$X hits zero, when the bivariate process $$(X,L)$$(X,L) starts from a coordinate axis. This extends classical formulæ  by McKean (J Math Kyoto Univ 2:227–235, 1963) and Gor’kov (Soviet. Math. Dokl. 16:904–908, 1975) for integrated Brownian motion.