Optimal Filtering For Partially Observed Point Processes Using Trans-Dimensional Sequential Monte Carlo

Continuous-time marked point processes appear in many areas of science and engineering including queuing theory, seismology, neuroscience and finance. In numerous applications, these point processes are unobserved but actually drive an observation process. Here, we are interested in optimal sequential Bayesian estimation of such partially observed point processes. This class of filtering problems is non-standard as there is typically no underlying Markov structure and the likelihood function relating the observations to the point process has a complex form. Hence, except in very specific cases it is impossible to solve them in closed-form. We develop an original trans-dimensional Sequential Monte Carlo method to address this class of problems. An application to partially observed queues is presented.