Upper first-exit times of compound poisson processes revisited

Abstract

For a compound Poisson process (CPP) with only positive jumps, an elegant formula connects the density of the hitting time for a lower straight line with that of the process itself at time t, h (x; t), considered as a function of time and position jointly. We prove an analogous (albeit more complicated) result for the first time the CPP crosses an upper straight line. We also consider the conditional density of the CPP at time t, given that the upper line has not been reached before t. Finally, it is shown how to compute certain moment integrals of h.