Poissonian occupation times of refracted Lévy processes with applications

Abstract

. Inspired by the work of Lkabous (2021), we consider Poissonian occupation times below level 0 of a refracted Lévy process where its premium rate is adaptive. In this model, occupation time is accumulated once the surplus process is observed to be negative at Poisson arrival times. Our analysis depends on various exit identities for refracted Lévy processes observed at Poisson arrival times. As an application of Poissonian occupation times, we derive an explicit expression for the probability of Parisian ruin with Erlang(2, λ) implementation delays. The resulting main quantities are in terms of scale functions.