A recursive method for computing moments of Hawkes intensities: application to the potential approach of credit risk

This paper explores an alternative to the structural models and reduced models in credit risk. The approach we use is called the potential approach. In the context of credit risk, it consists in assuming that the survival probability of a company is equal to the ratio of the expected value of a supermartingale divided by its initial value. This approach, that was previously used for modelling the term structure of interest rates, is extended by the use of a self-exciting processess that is time-changed by the inverse of an alpha-stable subordinator. We derive a new recursive method that allows to compute all the moments of a self-exciting process intensity. We show that this method can be used to approximate the survival probabilities in the potential aproach. More specifically, we prove that the approximation converges and we provide a bound on the numerical error. Finally, we calibrate the model and show that it allows to properly fit survival probability curves that are highly convex.