Advanced Catastrophic Risk Modeling in Insurance: A Fokker-Planck Equation Approach

Abstract

This paper presents an advanced model for assessing risks associated with potential catastrophic events faced by insurance companies. The model focuses on describing the behavior of claims related to phenomena that can have severe and far-reaching consequences. The mathematical foundation of this model is based on the Fokker-Planck equation, specifically its fractional form, which provides a robust framework for capturing the dynamics of risk processes. By modeling the solution to these equations, we derive the density function of the risk process, enabling a comprehensive understanding of the evolution of catastrophic events. The study emphasizes perturbed risk processes, utilizing fractional Brownian motion to model both normal and anomalous diffusion by varying the Hurst index. A key component of this approach is the calculation of the ruin probability, a critical risk measure in actuarial science, which is evaluated for a variety of models with corresponding numerical implementations. This approach offers a novel perspective on actuarial risk modeling, presenting a new methodology for coupling the severity of claims with the frequency of occurrence. The final fractional partial differential equations open a gate to using numerical methods in the field for extreme risk measurement and modeling of catastrophic or abnormal events.