The time at which a Lévy process creeps

We show that if a Lévy process (X_t)_t≥0 creeps then, as a function of u, the renewal function V(t, u) of the bivariate ascending ladder process (L^-1, H) is absolutely continuous on [0, ∞) and left differentiable on (0, ∞), and the left derivative at u is proportional to the (improper) distribution function of the time at which the process creeps over level u, where the constant of proportionality is d^-1_H, the reciprocal of the (positive) drift of H. This allows us to add the term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As an application, we derive a Laplace transform identity which generalises the second factorization identity. We also relate Doney and Kyprianou’s extension of Vigon’s équation amicale inversée to creeping. Some results concerning the ladder process of X, including the second factorization identity, continue to hold for a general bivariate subordinator, and are given in this generality.