A dynamic contagion process and an application to credit risk

We introduce a new point process, the dynamic contagion process, by gener- alising the Hawkes process and the Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamic contagion impact from endogenous and exoge- nous factors of the underlying system. We have systematically analysed the theoretical distributional properties of this new process, based on the piece- wise 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 the probability generating function of the point process have been de- rived. An explicit example of specified jumps with exponential distributions is also given. The object of this study is to produce a general mathemati- cal 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. We provide an application of this process to credit risk, and the simulation algorithm for further industrial implementation and statistical analysis.