A Martingale Characterization of Counting Processes with Application to Traffic Flow

摘要

In this report, various types of counting processes possessing a Poisson nature are considered. Explicit characterizations of these processes have been derived in terms of martingale representations decomposing the counting process into a 'predictable' and 'unpredictable' part, relative to the available information. The martingale characterization has a general structure and can be used in problems of estimation and control utilizing recent results in martingale theory. The main contribution of this paper lies in emphasizing the modelling aspects. It is shown that, in many applications, the intensity function of a counting process possesses a random nature: it is driven by the counting process itself or an associated 'outside' process. Examples of traffic flow with counting observations have been given.