  Continuous-time stochastic processes play an important role in the
description of random phenomena, it is therefore of prime interest to study
particular variables depending on their paths, like stopping time for example.
One approach consists in pointing out explicit expressions of the probability
distributions, an other approach is rather based on the numerical generation of
the random variables. We propose an algorithm in order to generate the first
passage time through a given level of a one-dimensional jump diffusion. This
process satisfies a stochastic differential equation driven by a Brownian
motion and subject to random shocks characterized by an independent Poisson
process. Our algorithm belongs to the family of rejection sampling procedures,
also called exact simulation in this context: the outcome of the algorithm and
the stopping time under consideration are identically distributed. It is based
on both the exact simulation of the diffusion at a given time and on the exact
simulation of first passage time for continuous diffusions. It is therefore
based on an extension of the algorithm introduced by Herrmann and Zucca [16] in
the continuous framework. The challenge here is to generate the exact position
of a continuous diffusion conditionally to the fact that the given level has
not been reached before. We present the construction of the algorithm and give
numerical illustrations, conditions on the recurrence of jump diffusions are
also discussed.
