Some monotonicity and dependence properties of self-exciting point processes

Abstract

Point processes on the positive real axis which are positively self-exciting in a sense expressed by their martingale dynamics are studied in this paper. It is shown that such processes can be realized as increasing mappings of Poisson processes and are therefore associated in appropriate manners. Some examples are presented, including Hawkes, renewal, Pólya-Lundberg, Markov dependent, semi-Markov, in addition to other point processes. As corollaries an extension of the Burton-Waymire association result and a solution of the Glasserman conjecture are obtained. Some results on dependence in stochastic processes of interest in queueing are given as a by product.