  We consider the first-crossing-time problem through a constant boundary for a
Wiener process perturbed by random jumps driven by a counting process. On the
base of a sample-path analysis of the jump-diffusion process we obtain explicit
lower bounds for the first-crossing-time density and for the
first-crossing-time distribution function. In the case of the distribution
function, the bound is improved by use of processes comparison based on the
usual stochastic order. The special case of constant jumps driven by a Poisson
process is thoroughly discussed.
