The moment generating function of the stopping time for a linearly stopped Poisson process

First-crossing times of linear boundaries by Poisson processes are considered. In particular, the moment generating functions of stopping times of the form T = inf {t ≥ 0 \ X_t - ct ∉ (A, B)} are examined, where {X_t; t ≥ 0} is a homogeneous Poisson counting process and -∞ < A < B ≤ ∞. Exact and asymptotic (in B) expressions are obtained by analyzing certain delay-differential equations derived for this moment generating function by Dvoretsky, Kiefer and Wolfowitz. Further results are obtained for the case of a single, lower boundary (i.e., B = ∞) by exploiting results of Zacks who gives a series representation for the stopping-time distribution in this case. These results are used to obtain a closed-form expression for the cumulants of the stopping time in this latter case.