Hitting Numbers and Probabilities for Lévy Processes with One-Sided Jumps

We present some exact formulas for hitting probabilities and hitting numbers of one or several levels for a Lévy process {Xt | t ≥ 0} with no negative jumps, starting at zero. Let Nx be the number of visits of level x ∈ R and define τx = inf{t > 0 | Xt = x}. We determine the Laplace transform of f(x) = P[τx < ∞], x > 0. The counting variable Nx is finite if and only if EX1 6= 0 and the paths of Xt have locally bounded variation almost surely; in this case we derive the generating function h(ξ1, ξ2) = Eξ Nx1 1 ξ Nx2 2 , 0 ≤ ξ1, ξ2 ≤ 1, for two levels x2 > x1.