Persistence of integrated stable processes

Christophe Profeta (1) , Thomas Simon (2, 3)

1 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques

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 Z. Shi (2003). Along the way, we investigate the law of the stable process L evaluated at the first time its integral X hits zero, when the bivariate process (X,L) starts from a coordinate axis. This extends classical formulae by McKean (1963) and Gor'kov (1975) for integrated Brownian motion.

Keywords: Integrated process; Half-Cauchy distribution; Hitting place; Lower tail probability; Mellin transform; Persistence; Stable Lévy process

Domains: Probability [math.PR]

Christophe Profeta, Thomas Simon. Persistence of integrated stable processes. Probability Theory and Related Fields, 2015, 162 (3), pp.463-485. ⟨hal-00955712⟩