  We propose a numerical method to approximate the value function for the
optimal stopping problem of a piecewise deterministic Markov process (PDMP).
Our approach is based on quantization of the post jump location---inter-arrival
time Markov chain naturally embedded in the PDMP, and path-adapted time
discretization grids. It allows us to derive bounds for the convergence rate of
the algorithm and to provide a computable $\epsilon$-optimal stopping time. The
paper is illustrated by a numerical example.
