Thomas PEYRAT - A Multivariate Self-Exciting Processes with Dependencies for actuarial applications

The compound Poisson process is commonly used to model the loss process associated to a certain risk. This process relies on the assumption of independence between the counting process (a Poisson process) and the claim sizes (independent and identically distributed random variables), making the calculation of the expectation and correlation straightforward. However, these assumptions limit its applicability to more complex risk structures. To overcome these limitations, we propose a similar framework in which the counting process is replaced by a self-exciting process whose intensity is influenced by the amount of the claims. Introducing dependency between the counting process and the claims, however, induces theoretical challenges in the computations of the first two moments of the loss process. To this end, we introduce the class of Multidimensional Self-Exciting Processes with Dependencies (MSPD), for which we derive closed-form expressions for the expectation and correlation.