On the first passage of one-sided Lévy motions

We study the first passage problem for one-sided Lévy motions: how does such a motion, started at the origin, cross a barrier positioned at the point x (x>0)? Since one-sided Lévy motions are pure-jump processes, they always ‘leap’ over barriers (rather than crossing them continuously). We hence explore the following issues: (i) first passage times (FPTs)—how long would it take the motion to cross the barrier; (ii) first passage leapovers (FPLs)—how far would the motion leap over the barrier; and, (iii) what is dependence between the FPTs and the FPLs. Formulae for the joint Laplace transforms of the FPTs and FPLs are derived, and the scaling limits of the FPTs and FPLs are computed. The scaling limits turn out to display dramatically different behavior when: (i) the underlying Lévy motion has finite cumulants; and when, (ii) the underlying Lévy motion is heavy-tailed. Special attention is devoted to the investigation of the first passage problem for self-similar one-sided Lévy motions.