Asymptotic results for dynamic contagion processes with different exciting functions and application to risk models

A class of point processes is introduced, the so-called dynamic contagion processes having different exciting functions. This is a generalization of that of Hawkes processes as well as of Cox processes with Poisson shot-noise intensity. To define this class the cluster form representation is considered in a way such that the intensity function captures both the self-excited and externally excited jumps by using different exciting functions, that allows us to describe different generations of offspring. For this generalized class, we investigate some asymptotic behaviors such as the Law of Large Numbers, the Central Limit Theorem and the Large Deviation Principle. An application associated with risk models is also discussed under the assumption that the dynamics of contagion claims arrivals have different exciting functions.