Bayesian Inference for Partially Observed Multiplicative Intensity Processes

Poisson processes are used in various applications. In their homogeneous version, the intensity process is a deterministic constant whereas it depends on time in their inhomogeneous version. To allow for an endogenous evolution of the intensity process, we consider multiplicative intensity processes. Inference methods for such processes have been developed when the trajectories are fully observed, that is to say, when both the sizes of the jumps and the jumps instants are observed. In this paper, we deal with the case of a partially observed process: we assume that the jumps sizes are non- or partially observed whereas the time events are fully observed. Moreover, we consider the case where the initial state of the process at time 0 is unknown. The inference being strongly influenced by this quantity, we propose a sensible prior distribution on the initial state, using the probabilistic properties of the process. We illustrate the performances of our methodology on a large simulation study.