A recent application of point processes has emerged from the electronic trading of financial assets. Many securities are now traded on purely electronic exchanges where demand and supply are aggregated in limit order books. Buy and sell trades in the asset as well as quote additions and cancellations can then be interpreted as events that not only determine the shape of the order book, but also define point processes that exhibit a rich internal structure. A large class of such point processes are those driven by a diffusive intensity process. A flexible choice with favourable analytic properties is a Cox-Ingersoll-Ross (CIR) diffusion. We adopt a Bayesian perspective on the statistical inference for these doubly stochastic processes, and focus on filtering the latent intensity process. We derive analytic results for the moment generating function of its posterior distribution. This is achieved by solving a partial differential equation for a linearised version of the filtering equation. We also establish an efficient and simple numerical evaluation of the posterior mean and variance of the intensity process. This relies on extending an equivalence result between a point process with CIR-intensity and a partially observed population process. We apply these results to empirical datasets from foreign exchange trading. One objective is to assess whether a CIR-driven point process is a satisfactory model for the variations in trading activity. This is answered in the negative, as sudden bursts of activity impair the fit of any diffusive intensity model. Controlling for such spikes, we conclude with a discussion of the stochastic control of a market making strategy when the only information available are the times of buy and sell trades.