Weibull Distribution and Approximation, by the Finite Volume Method, of the Ultim Ruin Probability Constructed from the Hawkes Variable Memory Process

It measures the risk that a system or company fails to maintain its elf over time. In this article, we provide an approximation of the probability of ruin at the infinite horizon whose inter-arrivals of claims follow the Hawks process and the amount of claims follows the Weibull distribution, with independence between these two processes. Using the Finite Volume Method is a numerical approach for solving partial differential equations. It consists of dividing the computational domain into discrete volumes and applying local approximations to obtain a global solution. This method can be used to estimate complex probabilities., a stochastic model with variable memory, it is possible to capture the temporal dependence of events. This allows us to analyze situations where the past directly influences the probability of occurrence of future events. This approximation is done using the finite volume method, which is a numerical approach for solving partial differential equations. It consists of dividing the computational domain into discrete volumes and applying local approximations to obtain a global solution. This method can be used to estimate complex probabilities. This is the case in our work; which consists of solving a second-order integro-differential equation, two cases of which are considered on the Weibull parameter <i>η</i>: if <i>η</i>=1, then the distribution of claim amounts is exponential. On the other hand, if <i>η</i>≥2, then the results lead us to a system of linear equations for which we use the finite volume method to obtain a numerical solution.