Stopped compound Poisson process and related distributions

This chapter considers the first-crossing problem of a compound Poisson process with positive integer-valued jumps in a nondecreasing lower boundary. The cases where the boundary is a given linear function, a standard renewal process, or an arbitrary deterministic function are successively examined. Our interest is focused on the exact distribution of the first-crossing level (or time) of the compound Poisson process. It is shown that, in all cases, this law has a simple remarkable form which relies on an underlying polynomial structure. The impact of a raise of a lower deterministic boundary is also discussed.