Convolution equivalent lévy processes and first passage times

We investigate the behavior of Lévy processes with convolution equivalent Lévy measures, up to the time of first passage over a high level u. Such problems arise naturally in the context of insurance risk where u is the initial reserve. We obtain a precise asymptotic estimate on the probability of first passage occurring by time T . This result is then used to study the process conditioned on first passage by time T . The existence of a limiting process as uis demonstrated, which leads to precise estimates for the probability of other events relating to first passage, such as the overshoot. A discussion of these results, as they relate to insurance risk, is also given.