Abstract: Our article relates to the field of actuarial science, where the analysis of the probability of ruin is a fundamental issue for insurance companies. The stability of reserves is a key factor in ensuring the sustainability of insurance companies, and understanding the mechanisms that influence this risk allows for the optimization of management strategies. The main objective of this study is to establish an expression to calculate the probability of ruin over a finite time horizon. We use the Hawkes process to model the dynamics of claims arrivals, and we introduce Brownian motion at the level of reserve R(t) to incorporate unexpected variations in compensations. By adopting the assumption that the arrival of claims and their amounts, which follow an exponential distribution, are independent. Then, considering the claims modeled by α-stable distribution. The key ideas developed in this article are based on several aspects: The Hawkes process is used to describe the frequency of claims, taking into account the impact of past events on the future dynamics of losses. A stochastic oscillation (Brownian motion) is integrated into the model to reflect variations in the financial reserve. With the previous elements, a mathematical expression for the probability of ruin in a finite time is formulated to assess the level of risk that a reserve faces over a given period.Abstract: Our article relates to the field of actuarial science, where the analysis of the probability of ruin is a fundamental issue for insurance companies. The stability of reserves is a key factor in ensuring the sustainability of insurance companies, and understanding the mechanisms that influence this risk allows for the optimization of management stra...Show More