On a first-passage-time problem for the compound power-law process

We consider a first-passage-time problem for a compound Poisson process characterized by independent, identically and exponentially distributed jumps, occurring according to the power-law process. First of all, we refer to the conditional product moments of arrival times and to the interarrival times density of a power-law process. We then obtain the probability density of the crossing time through a linear boundary at the occurrence of the n-th jump. In particular, we express the first-passage-time density in terms of a conditional expectation involving the arrival times.