  We present an exact sampling method for the first passage event of a Levy
process. The idea is to embed the process into another one whose first passage
event can be sampled exactly, and then recover the part belonging to the former
from the latter. The method is based on several distributional properties that
appear to be new. We obtain general procedures to sample the first passage
event of a subordinator across a regular non-increasing boundary, and that of a
process with infinite Levy measure, bounded variation, and suitable drift
across a constant level or interval. We give examples of application to a
rather wide variety of Levy measures.
