The lower tail problem for homogeneous functionals of stable processes with no negative jumps

Thomas Simon (1)

1 Département de Mathématiques (Evry)

Let 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.

Keywords: Fluctuating additive functional; lower tail probability; self-similar process; stable process; Wiener-Hopf factorization in two dimensions.; Wiener-Hopf factorization in two dimensions

Domains: Probability [math.PR]

Thomas Simon. The lower tail problem for homogeneous functionals of stable processes with no negative jumps. 2007. ⟨hal-00126032v2⟩