报告题目：Towards a Comprehensive Understanding of Ruin Theory: The Role of Fractional Differential Equations in Risk Modelling
报告摘要：This research presentation offers a novel exploration in the field of ruin theory, with a specialized focus on the application of fractional differential equations. We commence with the introduction of a unique class of fractional differential operators. These operators form the backbone for a family of random variables, where the density functions are solutions to the fractional differential equations proposed.
Our methodology allows us to construct fractional integro-differential equations to calculate ruin probabilities in various risk models, with the inter-arrival time distributions from the aforementioned family of random variables. We specifically delve into Gamma-time risk models and fractional Poisson risk models, providing explicit solutions for ruin probabilities when the claim size distributions have rational Laplace transforms.
As a practical extension to this theoretical framework, we examine a complex Erlang risk model. The uniqueness of this model lies in its dynamic adjustment of the premium rate and claim size distribution, determined by the inter-arrival time and an independent random time window. In this context, the ruin probabilities comply with a system of fractional integro-differential equations. For a specific class of claim size distributions, this system is further convertible into a fractional differential equation system.
Our findings include explicit solutions for these fractional boundary problems, supported by various numerical examples. This study thus not only delves into the theoretical aspects of ruin theory and fractional differential equations but also successfully demonstrates their practical applications in more sophisticated risk models.