On exact sampling of the first passage event of a Lévy process with infinite Lévy measure and bounded variation

Exact sampling of the first passage event (FPE) of a Lévy process with infinite Lévy measure is challenging due to lack of analytic formulas. We present an approach to the sampling for processes with bounded variation. The idea is to embed a process for which we wish to sample the FPE into another process whose FPE can be sampled based on analytic formulas, and once the latter FPE is sampled, extract from it the part belonging to the former process. We obtain general procedures to sample the FPE across a regular nonincreasing boundary or out of an interval. Concrete algorithms are given for two important classes of Lévy processes. The approach is based on distributional results that appear to be new.