  Piecewise deterministic Markov processes (PDMPs) are a class of stochastic
processes with applications in several fields of applied mathematics spanning
from mathematical modeling of physical phenomena to computational methods. A
PDMP is specified by three characteristic quantities: the deterministic motion,
the law of the random event times, and the jump kernels. The applicability of
PDMPs to real world scenarios is currently limited by the fact that these
processes can be simulated only when these three characteristics of the process
can be simulated exactly. In order to overcome this problem, we introduce
discretisation schemes for PDMPs which make their approximate simulation
possible. In particular, we design both first order and higher order schemes
that rely on approximations of one or more of the three characteristics. For
the proposed approximation schemes we study both pathwise convergence to the
continuous PDMP as the step size converges to zero and convergence in law to
the invariant measure of the PDMP in the long time limit. Moreover, we apply
our theoretical results to several PDMPs that arise from the computational
statistics and mathematical biology literature.
