  The filtering of a Markov diffusion process on a manifold from counting
process observations leads to `large' changes in the conditional distribution
upon an observed event, corresponding to a multiplication of the density by the
intensity function of the observation process. If that distribution is
represented by unweighted samples or particles, they need to be jointly
transformed such that they sample from the modified distribution. In previous
work, this transformation has been approximated by a translation of all the
particles by a common vector. However, such an operation is ill-defined on a
manifold, and on a vector space, a constant gain can lead to a wrong estimate
of the uncertainty over the hidden state. Here, taking inspiration from the
feedback particle filter (FPF), we derive an asymptotically exact filter
(called ppFPF) for point process observations, whose particles evolve according
to intrinsic (i.e. parametrization-invariant) dynamics that are composed of the
dynamics of the hidden state plus additional control terms. While not sharing
the gain-times-error structure of the FPF, the optimal control terms are
expressed as solutions to partial differential equations analogous to the
weighted Poisson equation for the gain of the FPF. The proposed filter can
therefore make use of existing approximation algorithms for solutions of
weighted Poisson equations.
