Ruin Probabilities in an Erlang Risk Model with Dependence Structure Based on an Independent Gamma-Distributed Time Window

In this paper, we investigate an Erlang risk model wherein the premium rate and claim size distribution are dynamically adjusted based on the inter-arrival time and an independent random time window. The ruin probabilities within this model adhere to a system of fractional integro-differential equations. For a specific class of claim size distributions, this system can be further transformed into a fractional differential equation system. We provide explicit solutions for these fractional boundary problems and illustrate our findings with several numerical examples