  This paper deals with the optimal stopping problem under partial observation
for piecewise-deterministic Markov processes. We first obtain a recursive
formulation of the optimal filter process and derive the dynamic programming
equation of the partially observed optimal stopping problem. Then, we propose a
numerical method, based on the quantization of the discrete-time filter process
and the inter-jump times, to approximate the value function and to compute an
actual $\epsilon$-optimal stopping time. We prove the convergence of the
algorithms and bound the rates of convergence.
