  The use of stochastic models, in effect piecewise deterministic Markov
processes (PDMP), has become increasingly popular especially for the modeling
of chemical reactions and cell biophysics. Yet, exact simulation methods, for
the simulation of these models in evolving environments, are limited by the
need to find the next jumping time at each recursion of the algorithm. Here, we
report on a new general method to find this jumping time for the True Jump
Method. It is based on an expression in terms of ordinary differential
equations for which efficient numerical methods are available. As such, our new
result makes it possible to study numerically stochastic models for which
analytical formulas are not available thereby providing a way to approximate
the state distribution for example. We conclude that the wide use of event
detection schemes for the simulation of PDMPs should be strongly reconsidered.
The only relevant remaining question being the efficiency of our method
compared to the Fictitious Jump Method, question which is strongly case
dependent.
