Approximation of the Ultim Ruin Probability by the Finite Difference Method of a Variable-memory Process (HAWKES process) With a Distribution of WEIBULL

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Résumé

In insurance risk management, the probability of ruin is a very important metric to assess. In this article, we give an approximation of the probability of ruin at the infinite horizon, where the inter-arrivals of claims follow the HAWKES process and the amount of claims follows the WEIBULL distribution, with independence between its two processes. This approximation is made using numerical analysis methods, it consists in solving a second-order integro-differential equation of which two cases are considered on the parameterηof WEIBULL: ifηis equal to 1, then the distribution of the amounts of claims is exponential, which brings us back to the risk model established in Badini et al. (2024). On the other hand, ifηis greater than 1, then the results lead us to a system of linear equations for which we use the finite di_erence method to obtain a numerical solution. This method is used in both cases (η = 1 andη> 1) for u ranging from 0 to 100, so we obtain the analytical solution.

Mots-clés

numerical method, integro-differential equation, probability of ruin, system of linear equations