Point processes with contagion and an application to credit risk

We introduce a new point process, the dynamic contagion process, by generalising the self excited Hawkes process (with exponential decay) by Hawkes (1971) and the Cox process with shot noise intensity by Dassios and Jang (2003). Our process includes both self excited and externally excited jumps, which can be used to model the dynamic contagion impact from endogenous and exogenous factors of the underlying system. We have systematically analysed the theoretical properties of this new process, based on the piecewise deterministic Markov process theory developed by Davis (1984), and the extension of the martingale methodology used by Dassios and Jang (2003). The analytic expressions of the Laplace transform of the intensity process and probability generating function of the point process have been derived. An explicit example of specified jumps with exponential distributions is also given. The object of this study is to produce a general mathematical framework for modelling the dependence structure of arriving events with dynamic contagion, which has the potential to be applicable to a variety of problems in economics, finance and insurance, such as credit risk and catastrophe risk. We provide an application of this process to credit risk, and the simulation algorithm for further industrial implementation and statistical analysis.