Continuous time filtering for a classo of marked doubly stochastic Poisson processes

We model a sequence of events by using a class of marked doubly stochastic Poisson processes where the intensity is given by a generalization of the classical shot noise process, specified as a positive function of another nonexplosive marked point process. To filter the unobservable intensity, a time recursion is constructed to characterize a sequence of filtering distributions, that is, the conditional distributions of the intensity, given the past observations, evaluated at opportunely chosen time instants. To approximate this sequence, we consider a discrete approximation with random support by implementing a particle filter, in which we draw recursively from each filtering distribution. In the case in which the pair formed by the marked point process and by the intensity is a Markov process, this filtering recursion can be related to the classical filtering theory.