  In the first part of this paper we study approximations of trajectories of
Piecewise Deter-ministic Processes (PDP) when the flow is not explicit by the
thinning method. We also establish a strong error estimate for PDPs as well as
a weak error expansion for Piecewise Deterministic Markov Processes (PDMP).
These estimates are the building blocks of the Multilevel Monte Carlo (MLMC)
method which we study in the second part. The coupling required by MLMC is
based on the thinning procedure. In the third part we apply these results to a
2-dimensional Morris-Lecar model with stochastic ion channels. In the range of
our simulations the MLMC estimator outperforms the classical Monte Carlo one.
