  In this paper, we present a novel methodology to perform Bayesian inference
for Cox processes in which the intensity function is driven by a diffusion
process. The novelty lies in the fact that no discretization error is involved,
despite the non-tractability of both the likelihood function and the transition
density of the diffusion. The methodology is based on an MCMC algorithm and its
exactness is built on retrospective sampling techniques. The efficiency of the
methodology is investigated in some simulated examples and its applicability is
illustrated in some real data analyzes.
