Extension of the Risk Model From a Hawkes Variable Memory Process via the Spearman Copula

The ultimate ruin probability of an insurance company throughout its operating life remains and continues to be a major and very complex concern for the latter. Although this probability of ruin can be modeled using stochastic processes, its determination remains particularly complex. This mathematical complexity represents a considerable obstacle for researchers in this field. Previous research has mainly focused on risk models built from Markovian processes. In this manuscript, we focus on risk models derived from non-Markovian processes, those whose claim arrivals are governed by variable memory processes, in particular Hawkes processes, and depend on the claim amount which follows an exponential distribution using Spearman’s copula. This allowed us to obtain a simplified expression of the ultimate probability of ruin and to carry out simulations. Before getting into the heart of the subject, we provide an extensive literature review on variable memory counting processes. This includes an in-depth exploration of Hawkes processes, copulas, and infinite time horizon ruin probabilities. In addition, we present the necessary mathematical tools essential for understanding the content of this article, especially designed for people with a fundamental understanding of actuarial science.