A dynamic contagion process for modelling contagion risk in finance and insurance

We introduce a new point process, the dynamic contagion process, by generalising the Hawkes process and Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamics of contagion impact from endogenous and exogenous factors of the underlying system. We systematically analyse the theoretical distributional properties of this new process, based on the piecewise-deterministic Markov process theory developed in Davis (1984), and the extension of the martingale methodology used in Dassios and Embrechts (1989). The analytic expressions of the Laplace transform of the intensity process and probability generating function of the point process are derived. A simulation algorithm is provided for further industrial implementation and statistical analysis. Some extensions of this process and comparison with other similar processes are also investigated. The major 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. We apply our research to the default probability of credit risk and ruin probability of risk theory