On the range of a Lévy risk process with fair valuation of insurance contracts

In the context of insurance risk management, large fluctuations in the surplus process represent a critical source of risk, with implications for the financial stability and resilience of insurers. Understanding and quantifying such variability is therefore essential for assessing the financial robustness of insurers. In this paper, we investigate the range of a Lévy risk process, which captures surplus variability by tracking the difference between the running supremum and infimum within a given time horizon. We derive new fluctuation identities for the inverse range time under both continuous and Poissonian observation schemes, extending results that were previously available only for Brownian motion in the existing literature. In addition, we study the joint Laplace transform of the inverse range time and the previous extremum time. For illustration, explicit expressions are obtained for the Brownian risk process and the Cramér–Lundberg risk model with exponential claims. Finally, we apply our results to the fair valuation of insurance contracts associated with the range process.