Abstract
Let us consider a sequence of events characterized by the values assumed by a certain quantity of interest and by the times at which they took place. Data of this kind can be naturally modelled as marked point processes (MPP) which can be seen as sequences of random times where each time is complemented with a random vector (the mark) taking values in some measurable space. An important subclass of these processes is the class of marked doubly stochastic Poisson processes (DSPP) which are characterized by having the number of events in any given time interval to be conditionally Poisson distributed, given another positive stochastic process called intensity (see Last and Brandt (1995)). We consider a class of marked DSPP in which the intensity process is a given function of a non-explosive positive jump process that we characterize through the distributions of jump times and sizes. A central problem faced when modelling with these processes is the filtering of the underlying, and typically unobservable, intensity. To solve this problem, a time recursion is constructed, to characterize the filtering distribution at certain time instants in terms of past filtering distributions. To approximate the filter, an algorithm can be constructed, in which samples are drawn recursively from each filtering distribution. These samples can be viewed, in the optic of particle filters, as discrete approximations with random support (the particles) of the distributions of interest. Under some technical conditions, a central limit theorem can be proved.