Probability of ruin within finite time and Cramér–Lundberg inequality for fractional risk processes

Abstract While the interarrival times of the classical Poisson process are exponentially distributed, complex systems often exhibit non-exponential patterns, motivating the use of the fractional Poisson process, in which interarrival times follow a Mittag–Leffler distribution. This paper investigates the associated risk process, describes its Cramér–Lundberg formula and establishes a relationship between the continuous premium rate and the fractional claim frequency. For a compound fractional risk process with exponential claims, we derive closed-form expressions for the finite-time ruin probability. Furthermore, for a general claim distribution, we provide ruin probability estimates that can serve as a basis for developing reinsurance strategies.