Exit identities for Lévy processes observed at Poisson arrival times

Abstract

For a spectrally one-sided Lévy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this type for processes reflected either from above or from below. The resulting Laplace transforms of the main quantities of interest are in terms of scale functions and turn out to be simple analogues of the classical formulas.

n-Dimensional Laplace Transforms of Occupation Times for Spectrally Negative Lévy Processes

“…We only prove identity (10). Applying definition (6), changing the order of integrals, and then, by identity (2), we have…”

“…Using techniques developed in Albrecher et al [6], in Li and Zhou [7], a Poisson approach is adopted to find joint Laplace transforms for occupation times over two disjoint intervals for general spectrally negative Lévy processes. This approach uses a property of Poisson random measure and can be effectively implemented.…”

Lévy insurance risk process with Poissonian taxation

“…n = 1) is known to result in nice explicit formulas (see Albrecher, Cheung et al (2011, Sections 2 and 4.1); Albrecher et al (2013, Section 2)). Since then, ruin theory under a Poissonian observer has been further developed by Albrecher & Ivanovs (2013) and Albrecher et al (2015), who looked at a Markov additive risk process and a Lévy risk process, respectively. Indeed, exponential inter-observation times are also related to the case of constant bankruptcy rate in the (Gamma-)Omega risk model.…”

A note on Parisian ruin under a hybrid observation scheme

“…Remark 2. It is interesting to note that the function Θ (q) (x; r, λ) is expressed in terms of scale functions Λ (q) (x; r, λ), Λ (q) (x; r) and Z q (x, Φ λ+q ) related to Parisian ruins (20), (1) and (3) respectively. It could be called the hybrid scale function.…”

“…Recently, Poisson observations problems have attracted considerable attention. In this case, the risk process is monitored discretely at arrival epochs of an independent Poisson process, which can be interpreted as the observation times of the regulatory body, see Albrecher et al [1] and Li et al [9] among others. A new definition of ruin has been studied in a spectrally negative Lévy setup by Li et al [9].…”